diff --git a/doc/Articles/Autodiff/17-468.bbl b/doc/Articles/Autodiff/17-468.bbl new file mode 100644 index 000000000..c63d3c847 --- /dev/null +++ b/doc/Articles/Autodiff/17-468.bbl @@ -0,0 +1,1475 @@ +\begin{thebibliography}{208} +\providecommand{\natexlab}[1]{#1} +\providecommand{\url}[1]{\texttt{#1}} +\expandafter\ifx\csname urlstyle\endcsname\relax + \providecommand{\doi}[1]{doi: #1}\else + \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi + +\bibitem[Abadi et~al.(2016)Abadi, Agarwal, Barham, Brevdo, Chen, Citro, + Corrado, Davis, Dean, Devin, et~al.]{abadi2016tensorflow} +Mart{\'\i}n Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, + Craig Citro, Greg~S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et~al. +\newblock {TensorFlow}: Large-scale machine learning on heterogeneous + distributed systems. +\newblock \emph{arXiv preprint arXiv:1603.04467}, 2016. + +\bibitem[Adamson and Winant(1969)]{Adamson1969} +D.~S. Adamson and C.~W. Winant. +\newblock A {SLANG} simulation of an initially strong shock wave downstream of + an infinite area change. +\newblock In \emph{Proceedings of the Conference on Applications of + Continuous-System Simulation Languages}, pages 231--40, 1969. + +\bibitem[Agarwal et~al.(2016)Agarwal, Bullins, and Hazan]{agarwal2016second} +Naman Agarwal, Brian Bullins, and Elad Hazan. +\newblock Second order stochastic optimization in linear time. +\newblock Technical Report arXiv:1602.03943, arXiv preprint, 2016. + +\bibitem[{Al Seyab} and Cao(2008)]{AlSeyab2008} +R.~K. {Al Seyab} and Y.~Cao. +\newblock Nonlinear system identification for predictive control using + continuous time recurrent neural networks and automatic differentiation. +\newblock \emph{Journal of Process Control}, 18\penalty0 (6):\penalty0 + 568--581, 2008. +\newblock \doi{10.1016/j.jprocont.2007.10.012}. + +\bibitem[Amos and Kolter(2017)]{amos2017optnet} +Brandon Amos and J~Zico Kolter. +\newblock {OptNet}: Differentiable optimization as a layer in neural networks. +\newblock \emph{arXiv preprint arXiv:1703.00443}, 2017. + +\bibitem[Apostolopoulou et~al.(2009)Apostolopoulou, Sotiropoulos, Livieris, and + Pintelas]{Apostolopoulou2009} +Marianna~S. Apostolopoulou, Dimitris~G. Sotiropoulos, Ioannis~E. Livieris, and + Panagiotis Pintelas. +\newblock A memoryless {BFGS} neural network training algorithm. +\newblock In \emph{7th IEEE International Conference on Industrial Informatics, + INDIN 2009}, pages 216--221, June 2009. +\newblock \doi{10.1109/INDIN.2009.5195806}. + +\bibitem[Appel(1989)]{appel1989runtime} +Andrew~W Appel. +\newblock Runtime tags aren't necessary. +\newblock \emph{Lisp and Symbolic Computation}, 2\penalty0 (2):\penalty0 + 153--162, 1989. + +\bibitem[Bahdanau et~al.(2014)Bahdanau, Cho, and Bengio]{bahdanau2014neural} +Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. +\newblock Neural machine translation by jointly learning to align and + translate. +\newblock \emph{arXiv preprint arXiv:1409.0473}, 2014. + +\bibitem[Barrett and Siskind(2013)]{Barrett2013} +Daniel~Paul Barrett and Jeffrey~Mark Siskind. +\newblock {Felzenszwalb-Baum-Welch}: Event detection by changing appearance. +\newblock \emph{arXiv preprint arXiv:1306.4746}, 2013. + +\bibitem[Bastien et~al.(2012)Bastien, Lamblin, Pascanu, Bergstra, Goodfellow, + Bergeron, Bouchard, Warde-Farley, and Bengio]{Bastien2012} +Frédéric Bastien, Pascal Lamblin, Razvan Pascanu, James Bergstra, Ian + Goodfellow, Arnaud Bergeron, Nicolas Bouchard, David Warde-Farley, and Yoshua + Bengio. +\newblock Theano: new features and speed improvements. +\newblock Deep Learning and Unsupervised Feature Learning NIPS 2012 Workshop, + 2012. + +\bibitem[Bauer(1974)]{Bauer1974} +Friedrich~L. Bauer. +\newblock Computational graphs and rounding error. +\newblock \emph{SIAM Journal on Numerical Analysis}, 11\penalty0 (1):\penalty0 + 87--96, 1974. + +\bibitem[Baydin et~al.(2016{\natexlab{a}})Baydin, Pearlmutter, and + Siskind]{baydin2016diffsharp} +Atılım~Güneş Baydin, Barak~A. Pearlmutter, and Jeffrey~Mark Siskind. +\newblock Diffsharp: An {AD} library for {.NET} languages. +\newblock In \emph{7th International Conference on Algorithmic Differentiation, + Christ Church Oxford, UK, September 12--15, 2016}, 2016{\natexlab{a}}. +\newblock Also arXiv:1611.03423. + +\bibitem[Baydin et~al.(2016{\natexlab{b}})Baydin, Pearlmutter, and + Siskind]{baydin2016tricks} +Atılım~Güneş Baydin, Barak~A. Pearlmutter, and Jeffrey~Mark Siskind. +\newblock Tricks from deep learning. +\newblock In \emph{7th International Conference on Algorithmic Differentiation, + Christ Church Oxford, UK, September 12--15, 2016}, 2016{\natexlab{b}}. +\newblock Also arXiv:1611.03777. + +\bibitem[Baydin et~al.(2018)Baydin, Cornish, Rubio, Schmidt, and + Wood]{baydin2017online} +Atılım~Güneş Baydin, Robert Cornish, David~Martínez Rubio, Mark Schmidt, + and Frank Wood. +\newblock Online learning rate adaptation with hypergradient descent. +\newblock In \emph{Sixth International Conference on Learning Representations + (ICLR), Vancouver, Canada, April 30--May 3, 2018}, 2018. + +\bibitem[Beda et~al.(1959)Beda, Korolev, Sukkikh, and Frolova]{Beda1959} +L.~M. Beda, L.~N. Korolev, N.~V. Sukkikh, and T.~S. Frolova. +\newblock Programs for automatic differentiation for the machine {BESM} (in + {Russian}). +\newblock Technical report, Institute for Precise Mechanics and Computation + Techniques, Academy of Science, Moscow, USSR, 1959. + +\bibitem[Bell and Burke(2008)]{Bell2008} +Bradley~M. Bell and James~V. Burke. +\newblock Algorithmic differentiation of implicit functions and optimal values. +\newblock In C.~H. Bischof, H.~M. Bücker, P.~Hovland, U.~Naumann, and J.~Utke, + editors, \emph{Advances in Automatic Differentiation}, volume~64 of + \emph{Lecture Notes in Computational Science and Engineering}, pages 67--77. + Springer Berlin Heidelberg, 2008. +\newblock \doi{10.1007/978-3-540-68942-3_7}. + +\bibitem[Bendtsen and Stauning(1996)]{Bendtsen1996} +Claus Bendtsen and Ole Stauning. +\newblock {FADBAD}, a flexible {C++} package for automatic differentiation. +\newblock Technical Report IMM-REP-1996-17, Department of Mathematical + Modelling, Technical University of Denmark, Lyngby, Denmark, 1996. + +\bibitem[Bengio et~al.(2013)Bengio, Courville, and + Vincent]{bengio2013representation} +Yoshua Bengio, Aaron Courville, and Pascal Vincent. +\newblock Representation learning: A review and new perspectives. +\newblock \emph{{IEEE} Transactions on Pattern Analysis and Machine + Intelligence}, 35\penalty0 (8):\penalty0 1798--1828, 2013. + +\bibitem[Bert and Malik(1996)]{Bert1996} +Charles~W. Bert and Moinuddin Malik. +\newblock Differential quadrature method in computational mechanics: A review. +\newblock \emph{Applied Mechanics Reviews}, 49, 1996. +\newblock \doi{10.1115/1.3101882}. + +\bibitem[Berz et~al.(1996)Berz, Makino, Shamseddine, Hoffstätter, and + Wan]{Berz1996} +Martin Berz, Kyoko Makino, Khodr Shamseddine, Georg~H. Hoffstätter, and Weishi + Wan. +\newblock {COSY} {INFINITY} and its applications in nonlinear dynamics. +\newblock In M.~Berz, C.~Bischof, G.~Corliss, and A.~Griewank, editors, + \emph{Computational Differentiation: Techniques, Applications, and Tools}, + pages 363--5. Society for Industrial and Applied Mathematics, Philadelphia, + PA, 1996. + +\bibitem[Bischof et~al.(1996)Bischof, Carle, Corliss, Griewank, and + Hovland]{Bischof1996} +Christian Bischof, Alan Carle, George Corliss, Andreas Griewank, and Paul + Hovland. +\newblock {ADIFOR} 2.0: Automatic differentiation of {Fortran} 77 programs. +\newblock \emph{Computational Science Engineering, IEEE}, 3\penalty0 + (3):\penalty0 18--32, 1996. +\newblock \doi{10.1109/99.537089}. + +\bibitem[Bischof et~al.(1997)Bischof, Roh, and {Mauer-Oats}]{Bischof1997} +Christian Bischof, Lucas Roh, and Andrew {Mauer-Oats}. +\newblock {ADIC}: An extensible automatic differentiation tool for {ANSI-C}. +\newblock \emph{Software Practice and Experience}, 27\penalty0 (12):\penalty0 + 1427--56, 1997. + +\bibitem[Bischof et~al.(2002)Bischof, Bücker, and Lang]{Bischof2002} +Christian~H. Bischof, H.~Martin Bücker, and Bruno Lang. +\newblock Automatic differentiation for computational finance. +\newblock In E.~J. Kontoghiorghes, B.~Rustem, and S.~Siokos, editors, + \emph{Computational Methods in Decision-Making, Economics and Finance}, + volume~74 of \emph{Applied Optimization}, pages 297--310. Springer US, 2002. +\newblock \doi{10.1007/978-1-4757-3613-7_15}. + +\bibitem[Bischof et~al.(2006)Bischof, Bücker, Rasch, Slusanschi, and + Lang]{Bischof2006} +Christian~H. Bischof, H.~Martin Bücker, Arno Rasch, Emil Slusanschi, and Bruno + Lang. +\newblock Automatic differentiation of the general-purpose computational fluid + dynamics package {FLUENT}. +\newblock \emph{Journal of Fluids Engineering}, 129\penalty0 (5):\penalty0 + 652--8, 2006. +\newblock \doi{10.1115/1.2720475}. + +\bibitem[Bischof et~al.(2008)Bischof, Hovland, and Norris]{Bischof2008} +Christian~H. Bischof, Paul~D. Hovland, and Boyana Norris. +\newblock On the implementation of automatic differentiation tools. +\newblock \emph{Higher-Order and Symbolic Computation}, 21\penalty0 + (3):\penalty0 311--31, 2008. +\newblock \doi{10.1007/s10990-008-9034-4}. + +\bibitem[Boltyanskii et~al.(1960)Boltyanskii, Gamkrelidze, and + Pontryagin]{Boltyanskii-Gamkrelidze-Pontryagin-1960a} +V.~G. Boltyanskii, R.~V. Gamkrelidze, and L.~S. Pontryagin. +\newblock The theory of optimal processes {I}: The maximum principle. +\newblock \emph{Izvest. Akad. Nauk S.S.S.R. Ser. Mat.}, 24:\penalty0 3--42, + 1960. + +\bibitem[Bottou(2010)]{bottou2010large} +L{\'e}on Bottou. +\newblock Large-scale machine learning with stochastic gradient descent. +\newblock In \emph{Proceedings of COMPSTAT'2010}, pages 177--186. Springer, + 2010. + +\bibitem[Bottou and {LeCun}(1988)]{bottou-lecun-88} +{L\'eon} Bottou and Yann {LeCun}. +\newblock {SN}: A simulator for connectionist models. +\newblock In \emph{Proceedings of NeuroNimes 88}, pages 371--382, Nimes, + France, 1988. +\newblock URL \url{http://leon.bottou.org/papers/bottou-lecun-88}. + +\bibitem[Bottou and LeCun(2002)]{LUSH2002} +L\'{e}on Bottou and Yann LeCun. +\newblock Lush reference manual, 2002. +\newblock URL \url{http://lush.sourceforge.net/doc.html}. + +\bibitem[Bottou et~al.(2016)Bottou, Curtis, and + Nocedal]{bottou2016optimization} +L{\'e}on Bottou, Frank~E. Curtis, and Jorge Nocedal. +\newblock Optimization methods for large-scale machine learning. +\newblock \emph{arXiv preprint arXiv:1606.04838}, 2016. + +\bibitem[Bottou(1998)]{Bottou1998} +Léon Bottou. +\newblock Online learning and stochastic approximations. +\newblock \emph{On-Line Learning in Neural Networks}, 17:\penalty0 9, 1998. + +\bibitem[Brezinski and Zaglia(1991)]{Brezinski1991} +Claude Brezinski and M.~Redivo Zaglia. +\newblock \emph{Extrapolation Methods: Theory and Practice}. +\newblock North-Holland, 1991. + +\bibitem[Bryson and Denham(1962)]{Bryson-1962a} +A.~E. Bryson and W.~F. Denham. +\newblock A steepest ascent method for solving optimum programming problems. +\newblock \emph{Journal of Applied Mechanics}, 29\penalty0 (2):\penalty0 247, + 1962. +\newblock \doi{10.1115/1.3640537}. + +\bibitem[Bryson and Ho(1969)]{Bryson-Ho-1969a} +Arthur~E. Bryson and Yu-Chi Ho. +\newblock \emph{Applied Optimal Control: Optimization, Estimation, and + Control}. +\newblock Blaisdell, Waltham, MA, 1969. + +\bibitem[Burden and Faires(2001)]{Burden2001} +Rirchard~L. Burden and J.~Douglas Faires. +\newblock \emph{Numerical Analysis}. +\newblock Brooks/Cole, 2001. + +\bibitem[Capriotti(2011)]{Capriotti2011} +Luca Capriotti. +\newblock Fast {Greeks} by algorithmic differentiation. +\newblock \emph{Journal of Computational Finance}, 14\penalty0 (3):\penalty0 3, + 2011. + +\bibitem[Carmichael and Sandu(1997)]{Carmichael1997} +Gregory~R. Carmichael and Adrian Sandu. +\newblock Sensitivity analysis for atmospheric chemistry models via automatic + differentiation. +\newblock \emph{Atmospheric Environment}, 31\penalty0 (3):\penalty0 475--89, + 1997. + +\bibitem[Carpenter et~al.(2015)Carpenter, Hoffman, Brubaker, Lee, Li, and + Betancourt]{carpenter2015stan} +Bob Carpenter, Matthew~D Hoffman, Marcus Brubaker, Daniel Lee, Peter Li, and + Michael Betancourt. +\newblock The {Stan} math library: Reverse-mode automatic differentiation in + {C++}. +\newblock \emph{arXiv preprint arXiv:1509.07164}, 2015. + +\bibitem[Carpenter et~al.(2016)Carpenter, Gelman, Hoffman, Lee, Goodrich, + Betancourt, Brubaker, Guo, Li, and Riddell]{carpenter2016stan} +Bob Carpenter, Andrew Gelman, Matt Hoffman, Daniel Lee, Ben Goodrich, Michael + Betancourt, Michael~A Brubaker, Jiqiang Guo, Peter Li, and Allen Riddell. +\newblock Stan: A probabilistic programming language. +\newblock \emph{Journal of Statistical Software}, 20:\penalty0 1--37, 2016. + +\bibitem[Casanova et~al.(2002)Casanova, Sharp, Final, Christianson, and + Symonds]{casanova2002application} +Daniele Casanova, Robin~S. Sharp, Mark Final, Bruce Christianson, and Pat + Symonds. +\newblock Application of automatic diffentiation to race car performance + optimisation. +\newblock In George Corliss, Christ\`{e}le Faure, Andreas Griewank, Lauren + Hasco\"{e}t, and Uwe Naumann, editors, \emph{Automatic Differentiation of + Algorithms}, pages 117--124. Springer-Verlag New York, Inc., New York, NY, + USA, 2002. +\newblock ISBN 0-387-95305-1. + +\bibitem[Charpentier and Ghemires(2000)]{Charpentier2000} +Isabelle Charpentier and Mohammed Ghemires. +\newblock Efficient adjoint derivatives: Application to the meteorological + model {Meso-NH}. +\newblock \emph{Optimization Methods and Software}, 13\penalty0 (1):\penalty0 + 35--63, 2000. + +\bibitem[Chen and Manning(2014)]{chen2014fast} +Danqi Chen and Christopher Manning. +\newblock A fast and accurate dependency parser using neural networks. +\newblock In \emph{Proceedings of the 2014 Conference on Empirical Methods in + Natural Language Processing (EMNLP)}, pages 740--750, 2014. + +\bibitem[Chetlur et~al.(2014)Chetlur, Woolley, Vandermersch, Cohen, Tran, + Catanzaro, and Shelhamer]{chetlur2014cudnn} +Sharan Chetlur, Cliff Woolley, Philippe Vandermersch, Jonathan Cohen, John + Tran, Bryan Catanzaro, and Evan Shelhamer. +\newblock {cuDNN}: Efficient primitives for deep learning. +\newblock \emph{arXiv preprint arXiv:1410.0759}, 2014. + +\bibitem[Chib and Greenberg(1995)]{Chib1995} +Siddhartha Chib and Edward Greenberg. +\newblock Understanding the {Metropolis-Hastings} algorithm. +\newblock \emph{The American Statistician}, 49\penalty0 (4):\penalty0 327--335, + 1995. +\newblock \doi{10.1080/00031305.1995.10476177}. + +\bibitem[Christianson(1994)]{christianson1994reverse} +Bruce Christianson. +\newblock Reverse accumulation and attractive fixed points. +\newblock \emph{Optimization Methods and Software}, 3\penalty0 (4):\penalty0 + 311--326, 1994. + +\bibitem[Christianson(2012)]{Christianson2012ALN} +Bruce Christianson. +\newblock A {L}eibniz notation for automatic differentiation. +\newblock In Shaun Forth, Paul Hovland, Eric Phipps, Jean Utke, and Andrea + Walther, editors, \emph{Recent Advances in Algorithmic Differentiation}, + volume~87 of \emph{Lecture Notes in Computational Science and Engineering}, + pages 1--9. Springer, Berlin, 2012. +\newblock ISBN 978-3-540-68935-5. +\newblock \doi{10.1007/978-3-642-30023-3_1}. + +\bibitem[Clifford(1873)]{Clifford1873} +William~K. Clifford. +\newblock Preliminary sketch of bi-quaternions. +\newblock \emph{Proceedings of the London Mathematical Society}, 4:\penalty0 + 381--95, 1873. + +\bibitem[Cohen and Molemaker(2009)]{cohen2009fast} +J~Cohen and M~Jeroen Molemaker. +\newblock A fast double precision cfd code using cuda. +\newblock \emph{Parallel Computational Fluid Dynamics: Recent Advances and + Future Directions}, pages 414--429, 2009. + +\bibitem[Collobert et~al.(2011)Collobert, Kavukcuoglu, and + Farabet]{collobert2011torch7} +Ronan Collobert, Koray Kavukcuoglu, and Cl{\'e}ment Farabet. +\newblock Torch7: A {Matlab}-like environment for machine learning. +\newblock In \emph{BigLearn, NIPS Workshop}, number EPFL-CONF-192376, 2011. + +\bibitem[Corliss(1988)]{Corliss1988} +George~F. Corliss. +\newblock \emph{Application of differentiation arithmetic}, volume~19 of + \emph{Perspectives in Computing}, pages 127--48. +\newblock Academic Press, Boston, 1988. + +\bibitem[Courbariaux et~al.(2015)Courbariaux, Bengio, and + David]{courbariaux2015binaryconnect} +Matthieu Courbariaux, Yoshua Bengio, and Jean-Pierre David. +\newblock Binaryconnect: Training deep neural networks with binary weights + during propagations. +\newblock In \emph{Advances in Neural Information Processing Systems}, pages + 3123--3131, 2015. + +\bibitem[Dalal and Triggs(2005)]{Dalal2005} +Navneet Dalal and Bill Triggs. +\newblock Histograms of oriented gradients for human detection. +\newblock In \emph{Proceedings of the 2005 IEEE Computer Society Conference on + Computer Vision and Pattern Recognition (CVPR'05)}, pages 886--93, + Washington, DC, USA, 2005. IEEE Computer Society. +\newblock \doi{10.1109/CVPR.2005.177}. + +\bibitem[Dauvergne and Hasco{\"e}t(2006)]{Dauvergne2006} +Benjamin Dauvergne and Laurent Hasco{\"e}t. +\newblock The data-flow equations of checkpointing in reverse automatic + differentiation. +\newblock In V.~N. Alexandrov, G.~D. {van Albada}, P.~M.~A. Sloot, and + J.~Dongarra, editors, \emph{Computational Science – ICCS 2006}, volume 3994 + of \emph{Lecture Notes in Computer Science}, pages 566--73, Dauvergne, 2006. + Springer Berlin. + +\bibitem[Dennis and Schnabel(1996)]{Dennis1996} +John~E. Dennis and Robert~B. Schnabel. +\newblock \emph{Numerical Methods for Unconstrained Optimization and Nonlinear + Equations}. +\newblock Classics in Applied Mathematics. Society for Industrial and Applied + Mathematics, Philadelphia, 1996. + +\bibitem[Dixon(1991)]{Dixon1991} +L.~C. Dixon. +\newblock Use of automatic differentiation for calculating {Hessians} and + {Newton} steps. +\newblock In A.~Griewank and G.~F. Corliss, editors, \emph{Automatic + Differentiation of Algorithms: Theory, Implementation, and Application}, + pages 114--125. SIAM, Philadelphia, PA, 1991. + +\bibitem[Duane et~al.(1987)Duane, Kennedy, Pendleton, and Roweth]{Duane1987} +Simon Duane, Anthony~D. Kennedy, Brian~J. Pendleton, and Duncan Roweth. +\newblock Hybrid {Monte Carlo}. +\newblock \emph{Physics Letters B}, 195\penalty0 (2):\penalty0 216--222, 1987. + +\bibitem[Duchi et~al.(2011)Duchi, Hazan, and Singer]{duchi2011adaptive} +John Duchi, Elad Hazan, and Yoram Singer. +\newblock Adaptive subgradient methods for online learning and stochastic + optimization. +\newblock \emph{Journal of Machine Learning Research}, 12\penalty0 + (Jul):\penalty0 2121--2159, 2011. + +\bibitem[Ekstr{\"o}m et~al.(2010)Ekstr{\"o}m, Visscher, Bast, Thorvaldsen, and + Ruud]{Ekstrom2010} +Ulf Ekstr{\"o}m, Lucas Visscher, Radovan Bast, Andreas~J. Thorvaldsen, and + Kenneth Ruud. +\newblock Arbitrary-order density functional response theory from automatic + differentiation. +\newblock \emph{Journal of Chemical Theory and Computation}, 6:\penalty0 + 1971--80, 2010. +\newblock \doi{10.1021/ct100117s}. + +\bibitem[Eriksson et~al.(1998)Eriksson, Gulliksson, Lindström, and Åke + Wedin]{Eriksson1998} +Jerry Eriksson, Mårten Gulliksson, Per Lindström, and Per Åke Wedin. +\newblock Regularization tools for training large feed-forward neural networks + using automatic differentiation. +\newblock \emph{Optimization Methods and Software}, 10\penalty0 (1):\penalty0 + 49--69, 1998. +\newblock \doi{10.1080/10556789808805701}. + +\bibitem[Eslami et~al.(2016)Eslami, Heess, Weber, Tassa, Szepesvari, + Kavukcuoglu, and Hinton]{eslami2016attend} +S.~M.~Ali Eslami, Nicolas Heess, Theophane Weber, Yuval Tassa, David + Szepesvari, Koray Kavukcuoglu, and Geoffrey~E. Hinton. +\newblock Attend, infer, repeat: Fast scene understanding with generative + models. +\newblock In D.~D. Lee, M.~Sugiyama, U.~V. Luxburg, I.~Guyon, and R.~Garnett, + editors, \emph{Advances in Neural Information Processing Systems 29}, pages + 3225--3233. Curran Associates, Inc., 2016. + +\bibitem[Finkel et~al.(2008)Finkel, Kleeman, and Manning]{Finkel2008} +Jenny~Rose Finkel, Alex Kleeman, and Christopher~D. Manning. +\newblock Efficient, feature-based, conditional random field parsing. +\newblock In \emph{Proceedings of the 46th Annual Meeting of the Association + for Computational Linguistics (ACL 2008)}, pages 959--67, 2008. + +\bibitem[Fornberg(1981)]{Fornberg1981} +Bengt Fornberg. +\newblock Numerical differentiation of analytic functions. +\newblock \emph{ACM Transactions on Mathematical Software}, 7\penalty0 + (4):\penalty0 512--26, 1981. +\newblock \doi{10.1145/355972.355979}. + +\bibitem[Forth(2006)]{Forth2006} +Shaun~A. Forth. +\newblock An efficient overloaded implementation of forward mode automatic + differentiation in {MATLAB}. +\newblock \emph{ACM Transactions on Mathematical Software}, 32\penalty0 + (2):\penalty0 195--222, 2006. + +\bibitem[Forth and Evans(2002)]{forth2002aerofoil} +Shaun~A. Forth and Trevor~P. Evans. +\newblock Aerofoil optimisation via {AD} of a multigrid cell-vertex {Euler} + flow solver. +\newblock In George Corliss, Christ{\`e}le Faure, Andreas Griewank, Laurent + Hasco{\"e}t, and Uwe Naumann, editors, \emph{Automatic Differentiation of + Algorithms: From Simulation to Optimization}, pages 153--160. Springer New + York, New York, NY, 2002. +\newblock ISBN 978-1-4613-0075-5. +\newblock \doi{10.1007/978-1-4613-0075-5_17}. + +\bibitem[Fourer et~al.(2002)Fourer, Gay, and Kernighan]{Fourer2002} +Robert Fourer, David~M. Gay, and Brian~W. Kernighan. +\newblock \emph{{AMPL}: A Modeling Language for Mathematical Programming}. +\newblock Duxbury Press, 2002. + +\bibitem[Gay(1996)]{Gay1996} +David~M. Gay. +\newblock Automatically finding and exploiting partially separable structure in + nonlinear programming problems. +\newblock Technical report, Bell Laboratories, Murray Hill, NJ, 1996. + +\bibitem[Gebremedhin et~al.(2009)Gebremedhin, Tarafdar, Pothen, and + Walther]{Gebremedhin2009} +Assefaw~H. Gebremedhin, Arijit Tarafdar, Alex Pothen, and Andrea Walther. +\newblock Efficient computation of sparse {Hessians} using coloring and + automatic differentiation. +\newblock \emph{INFORMS Journal on Computing}, 21\penalty0 (2):\penalty0 + 209--23, 2009. +\newblock \doi{10.1287/ijoc.1080.0286}. + +\bibitem[Gebremedhin et~al.(2013)Gebremedhin, Nguyen, Patwary, and + Pothen]{gebremedhin2013colpack} +Assefaw~H Gebremedhin, Duc Nguyen, Md~Mostofa~Ali Patwary, and Alex Pothen. +\newblock {ColPack}: Software for graph coloring and related problems in + scientific computing. +\newblock \emph{ACM Transactions on Mathematical Software (TOMS)}, 40\penalty0 + (1):\penalty0 1, 2013. + +\bibitem[Gershman and Goodman(2014)]{gershman2014amortized} +Samuel Gershman and Noah Goodman. +\newblock Amortized inference in probabilistic reasoning. +\newblock In \emph{Proceedings of the Annual Meeting of the Cognitive Science + Society}, number~36, 2014. + +\bibitem[Giering and Kaminski(1998)]{Giering1998} +Ralf Giering and Thomas Kaminski. +\newblock Recipes for adjoint code construction. +\newblock \emph{ACM Transactions on Mathematical Software}, 24:\penalty0 + 437--74, 1998. +\newblock \doi{10.1145/293686.293695}. + +\bibitem[Gimpel et~al.(2010)Gimpel, Das, and Smith]{Gimpel2010} +Kevin Gimpel, Dipanjan Das, and Noah~A. Smith. +\newblock Distributed asynchronous online learning for natural language + processing. +\newblock In \emph{Proceedings of the Fourteenth Conference on Computational + Natural Language Learning}, CoNLL '10, pages 213--222, Stroudsburg, PA, USA, + 2010. Association for Computational Linguistics. + +\bibitem[Girolami and Calderhead(2011)]{Girolami2011} +Mark Girolami and Be~Calderhead. +\newblock Riemann manifold {Langevin} and {Hamiltonian} {Monte Carlo} methods. +\newblock \emph{Journal of the Royal Statistical Society: Series B (Statistical + Methodology)}, 73\penalty0 (2):\penalty0 123--214, 2011. + +\bibitem[Goldberg(2016)]{goldberg2016primer} +Yoav Goldberg. +\newblock A primer on neural network models for natural language processing. +\newblock \emph{Journal of Artificial Intelligence Research}, 57:\penalty0 + 345--420, 2016. + +\bibitem[Goodfellow et~al.(2016)Goodfellow, Bengio, and + Courville]{goodfellow2016deep} +Ian Goodfellow, Yoshua Bengio, and Aaron Courville. +\newblock \emph{Deep Learning}. +\newblock MIT Press, 2016. +\newblock \url{http://www.deeplearningbook.org}. + +\bibitem[Gordon et~al.(2014)Gordon, Henzinger, Nori, and + Rajamani]{gordon2014probabilistic} +Andrew~D Gordon, Thomas~A Henzinger, Aditya~V Nori, and Sriram~K Rajamani. +\newblock Probabilistic programming. +\newblock In \emph{Proceedings of the on Future of Software Engineering}, pages + 167--181. ACM, 2014. + +\bibitem[Grabmeier and Kaltofen(2003)]{Grabmeier2003} +Johannes Grabmeier and Erich Kaltofen. +\newblock \emph{Computer Algebra Handbook: Foundations, Applications, Systems}. +\newblock Springer, 2003. + +\bibitem[Grabner et~al.(2008)Grabner, Pock, Gross, and Kainz]{Grabner2008} +Markus Grabner, Thomas Pock, Tobias Gross, and Bernhard Kainz. +\newblock Automatic differentiation for {GPU}-accelerated {2D/3D} registration. +\newblock In C.~H. Bischof, H.~M. Bücker, P.~Hovland, U.~Naumann, and J.~Utke, + editors, \emph{Advances in Automatic Differentiation}, volume~64 of + \emph{Lecture Notes in Computational Science and Engineering}, pages + 259--269. Springer Berlin Heidelberg, 2008. +\newblock \doi{10.1007/978-3-540-68942-3_23}. + +\bibitem[Grathwohl et~al.(2017)Grathwohl, Choi, Wu, Roeder, and + Duvenaud]{grathwohl2017backpropagation} +Will Grathwohl, Dami Choi, Yuhuai Wu, Geoff Roeder, and David Duvenaud. +\newblock Backpropagation through the void: Optimizing control variates for + black-box gradient estimation. +\newblock \emph{arXiv preprint arXiv:1711.00123}, 2017. + +\bibitem[Graves et~al.(2014)Graves, Wayne, and Danihelka]{graves2014neural} +Alex Graves, Greg Wayne, and Ivo Danihelka. +\newblock Neural {Turing} machines. +\newblock \emph{arXiv preprint arXiv:1410.5401}, 2014. + +\bibitem[Graves et~al.(2016)Graves, Wayne, Reynolds, Harley, Danihelka, + Grabska-Barwi{\'n}ska, Colmenarejo, Grefenstette, Ramalho, Agapiou, + et~al.]{graves2016hybrid} +Alex Graves, Greg Wayne, Malcolm Reynolds, Tim Harley, Ivo Danihelka, Agnieszka + Grabska-Barwi{\'n}ska, Sergio~G{\'o}mez Colmenarejo, Edward Grefenstette, + Tiago Ramalho, John Agapiou, et~al. +\newblock Hybrid computing using a neural network with dynamic external memory. +\newblock \emph{Nature}, 538\penalty0 (7626):\penalty0 471--476, 2016. + +\bibitem[Grefenstette et~al.(2015)Grefenstette, Hermann, Suleyman, and + Blunsom]{grefenstette2015learning} +Edward Grefenstette, Karl~Moritz Hermann, Mustafa Suleyman, and Phil Blunsom. +\newblock Learning to transduce with unbounded memory. +\newblock In \emph{Advances in Neural Information Processing Systems}, pages + 1828--1836, 2015. + +\bibitem[Griewank(1989)]{Griewank1989} +Andreas Griewank. +\newblock On automatic differentiation. +\newblock In M.~Iri and K.~Tanabe, editors, \emph{Mathematical Programming: + Recent Developments and Applications}, pages 83--108. Kluwer Academic + Publishers, 1989. + +\bibitem[Griewank(2003)]{Griewank2003} +Andreas Griewank. +\newblock A mathematical view of automatic differentiation. +\newblock \emph{Acta Numerica}, 12:\penalty0 321--98, 2003. +\newblock \doi{10.1017/S0962492902000132}. + +\bibitem[Griewank(2012)]{Griewank2012} +Andreas Griewank. +\newblock Who invented the reverse mode of differentiation? +\newblock \emph{Documenta Mathematica}, Extra Volume ISMP:\penalty0 389--400, + 2012. + +\bibitem[Griewank and Walther(2008)]{Griewank2008} +Andreas Griewank and Andrea Walther. +\newblock \emph{Evaluating Derivatives: Principles and Techniques of + Algorithmic Differentiation}. +\newblock Society for Industrial and Applied Mathematics, Philadelphia, 2008. +\newblock \doi{10.1137/1.9780898717761}. + +\bibitem[Griewank et~al.(2012)Griewank, Kulshreshtha, and + Walther]{griewank2012numerical} +Andreas Griewank, Kshitij Kulshreshtha, and Andrea Walther. +\newblock On the numerical stability of algorithmic differentiation. +\newblock \emph{Computing}, 94\penalty0 (2-4):\penalty0 125--149, 2012. + +\bibitem[Gruslys et~al.(2016)Gruslys, Munos, Danihelka, Lanctot, and + Graves]{gruslys2016memory} +Audrunas Gruslys, R{\'e}mi Munos, Ivo Danihelka, Marc Lanctot, and Alex Graves. +\newblock Memory-efficient backpropagation through time. +\newblock In \emph{Advances in Neural Information Processing Systems}, pages + 4125--4133, 2016. + +\bibitem[Gupta et~al.(2015)Gupta, Agrawal, Gopalakrishnan, and + Narayanan]{gupta2015deep} +Suyog Gupta, Ankur Agrawal, Kailash Gopalakrishnan, and Pritish Narayanan. +\newblock Deep learning with limited numerical precision. +\newblock In \emph{Proceedings of the 32nd International Conference on Machine + Learning (ICML-15)}, pages 1737--1746, 2015. + +\bibitem[Haase et~al.(2002)Haase, Langer, Lindner, and + M{\"u}hlhuber]{haase2002optimal} +Gundolf Haase, Ulrich Langer, Ewald Lindner, and Wolfram M{\"u}hlhuber. +\newblock Optimal sizing of industrial structural mechanics problems using + {AD}. +\newblock In \emph{Automatic Differentiation of Algorithms}, pages 181--188. + Springer, 2002. + +\bibitem[Hadjis et~al.(2015)Hadjis, Abuzaid, Zhang, and + R{\'e}]{hadjis2015caffe} +Stefan Hadjis, Firas Abuzaid, Ce~Zhang, and Christopher R{\'e}. +\newblock Caffe con troll: Shallow ideas to speed up deep learning. +\newblock In \emph{Proceedings of the Fourth Workshop on Data analytics in the + Cloud}, page~2. ACM, 2015. + +\bibitem[Hamilton(1837)]{Hamilton1837} +William~Rowan Hamilton. +\newblock Theory of conjugate functions, or algebraic couples; with a + preliminary and elementary essay on algebra as the science of pure time. +\newblock \emph{Transactions of the Royal Irish Academy}, 17:\penalty0 + 293--422, 1837. + +\bibitem[Hasco{\"e}t and Pascual(2013)]{Hascoet2013} +Laurent Hasco{\"e}t and Valérie Pascual. +\newblock The {Tapenade} automatic differentiation tool: principles, model, and + specification. +\newblock \emph{ACM Transactions on Mathematical Software}, 39\penalty0 (3), + 2013. +\newblock \doi{10.1145/2450153.2450158}. + +\bibitem[Hecht-Nielsen(1989)]{Hecht1989} +Robert Hecht-Nielsen. +\newblock Theory of the backpropagation neural network. +\newblock In \emph{International Joint Conference on Neural Networks, IJCNN + 1989}, pages 593--605. IEEE, 1989. + +\bibitem[Hinkins(1994)]{Hinkins1994} +Ruth~L. Hinkins. +\newblock Parallel computation of automatic differentiation applied to magnetic + field calculations. +\newblock Technical report, Lawrence Berkeley Lab., CA, 1994. + +\bibitem[Hinton and Ghahramani(1997)]{hinton1997generative} +Geoffrey~E. Hinton and Zoubin Ghahramani. +\newblock Generative models for discovering sparse distributed representations. +\newblock \emph{Philosophical Transactions of the Royal Society of London B: + Biological Sciences}, 352\penalty0 (1358):\penalty0 1177--1190, 1997. + +\bibitem[Hoffman and Gelman(2014)]{Hoffman2014} +Matthew~D. Hoffman and Andrew Gelman. +\newblock The no-{U}-turn sampler: Adaptively setting path lengths in + {H}amiltonian {M}onte {C}arlo. +\newblock \emph{Journal of Machine Learning Research}, 15:\penalty0 1351--1381, + 2014. + +\bibitem[Horn(1977)]{horn1977understanding} +Berthold K.~P. Horn. +\newblock Understanding image intensities. +\newblock \emph{Artificial Intelligence}, 8:\penalty0 201--231, 1977. + +\bibitem[Horwedel et~al.(1988)Horwedel, Worley, Oblow, and Pin]{Horwedel1988} +Jim~E. Horwedel, Brian~A. Worley, E.~M. Oblow, and F.~G. Pin. +\newblock {GRESS} version 1.0 user's manual. +\newblock Technical Memorandum ORNL/TM 10835, Martin Marietta Energy Systems, + Inc., Oak Ridge National Laboratory, Oak Ridge, 1988. + +\bibitem[Jerrell(1997)]{Jerrell1997} +Max~E. Jerrell. +\newblock Automatic differentiation and interval arithmetic for estimation of + disequilibrium models. +\newblock \emph{Computational Economics}, 10\penalty0 (3):\penalty0 295--316, + 1997. + +\bibitem[Jia et~al.(2014)Jia, Shelhamer, Donahue, Karayev, Long, Girshick, + Guadarrama, and Darrell]{jia2014caffe} +Yangqing Jia, Evan Shelhamer, Jeff Donahue, Sergey Karayev, Jonathan Long, Ross + Girshick, Sergio Guadarrama, and Trevor Darrell. +\newblock Caffe: Convolutional architecture for fast feature embedding. +\newblock In \emph{Proceedings of the 22nd ACM International Conference on + Multimedia}, pages 675--678. ACM, 2014. + +\bibitem[Johnson et~al.(2016)Johnson, Duvenaud, Wiltschko, Adams, and + Datta]{johnson2016composing} +Matthew Johnson, David~K Duvenaud, Alex Wiltschko, Ryan~P Adams, and Sandeep~R + Datta. +\newblock Composing graphical models with neural networks for structured + representations and fast inference. +\newblock In \emph{Advances in Neural Information Processing Systems}, pages + 2946--2954, 2016. + +\bibitem[Jones et~al.(1993{\natexlab{a}})Jones, Gomard, and + Sestoft]{jones1993partial} +Neil~D Jones, Carsten~K Gomard, and Peter Sestoft. +\newblock \emph{Partial evaluation and automatic program generation}. +\newblock Peter Sestoft, 1993{\natexlab{a}}. + +\bibitem[Jones and Launchbury(1991)]{jones1991unboxed} +Simon L~Peyton Jones and John Launchbury. +\newblock Unboxed values as first class citizens in a non-strict functional + language. +\newblock In \emph{Conference on Functional Programming Languages and Computer + Architecture}, pages 636--666. Springer, 1991. + +\bibitem[Jones et~al.(1993{\natexlab{b}})Jones, Hall, Hammond, Partain, and + Wadler]{jones1993glasgow} +SL~Peyton Jones, Cordy Hall, Kevin Hammond, Will Partain, and Philip Wadler. +\newblock The {G}lasgow {H}askell compiler: a technical overview. +\newblock In \emph{Proc. UK Joint Framework for Information Technology (JFIT) + Technical Conference}, volume~93, 1993{\natexlab{b}}. + +\bibitem[Joulin and Mikolov(2015)]{joulin2015inferring} +Armand Joulin and Tomas Mikolov. +\newblock Inferring algorithmic patterns with stack-augmented recurrent nets. +\newblock In \emph{Advances in Neural Information Processing Systems}, pages + 190--198, 2015. + +\bibitem[Juedes(1991)]{Juedes1991} +David~W. Juedes. +\newblock A taxonomy of automatic differentiation tools. +\newblock In A.~Griewank and G.~F. Corliss, editors, \emph{Automatic + Differentiation of Algorithms: Theory, Implementation, and Application}, + pages 315--29. Society for Industrial and Applied Mathematics, Philadelphia, + PA, 1991. + +\bibitem[Kingma and Ba(2015)]{kingma2015adam} +D.~Kingma and J.~Ba. +\newblock Adam: A method for stochastic optimization. +\newblock In \emph{The International Conference on Learning Representations + (ICLR), San Diego}, 2015. + +\bibitem[Kingma and Welling(2014)]{kingma2014auto} +Diederik~P. Kingma and Max Welling. +\newblock Auto-encoding variational {Bayes}. +\newblock In \emph{International Conference on Learning Representations}, 2014. + +\bibitem[Krizhevsky et~al.(2012)Krizhevsky, Sutskever, and + Hinton]{krizhevsky2012imagenet} +Alex Krizhevsky, Ilya Sutskever, and Geoffrey~E. Hinton. +\newblock {ImageNet} classification with deep convolutional neural networks. +\newblock In \emph{Advances in Neural Information Processing Systems}, pages + 1097--1105, 2012. + +\bibitem[Kubo and Iri(1990)]{Kubo1990} +K.~Kubo and M.~Iri. +\newblock {PADRE2}, version 1---user's manual. +\newblock Research Memorandum RMI 90-01, Department of Mathematical Engineering + and Information Physics, University of Tokyo, Tokyo, 1990. + +\bibitem[Kucukelbir et~al.(2017)Kucukelbir, Tran, Ranganath, Gelman, and + Blei]{kucukelbir2017automatic} +Alp Kucukelbir, Dustin Tran, Rajesh Ranganath, Andrew Gelman, and David~M. + Blei. +\newblock Automatic differentiation variational inference. +\newblock \emph{Journal of Machine Learning Research}, 18\penalty0 + (14):\penalty0 1--45, 2017. + +\bibitem[Kulkarni et~al.(2015)Kulkarni, Kohli, Tenenbaum, and + Mansinghka]{kulkarni2015picture} +Tejas~D. Kulkarni, Pushmeet Kohli, Joshua~B. Tenenbaum, and Vikash Mansinghka. +\newblock Picture: A probabilistic programming language for scene perception. +\newblock In \emph{The {IEEE} Conference on Computer Vision and Pattern + Recognition ({CVPR})}, June 2015. + +\bibitem[Kumar et~al.(2016)Kumar, Irsoy, Ondruska, Iyyer, Bradbury, Gulrajani, + Zhong, Paulus, and Socher]{pmlr-v48-kumar16} +Ankit Kumar, Ozan Irsoy, Peter Ondruska, Mohit Iyyer, James Bradbury, Ishaan + Gulrajani, Victor Zhong, Romain Paulus, and Richard Socher. +\newblock Ask me anything: Dynamic memory networks for natural language + processing. +\newblock In Maria~Florina Balcan and Kilian~Q. Weinberger, editors, + \emph{Proceedings of The 33rd International Conference on Machine Learning}, + volume~48 of \emph{Proceedings of Machine Learning Research}, pages + 1378--1387, New York, New York, USA, 20--22 Jun 2016. PMLR. + +\bibitem[Lawson(1971)]{Lawson1971} +C.~L. Lawson. +\newblock Computing derivatives using {W}-arithmetic and {U}-arithmetic. +\newblock Internal Computing Memorandum CM-286, Jet Propulsion Laboratory, + Pasadena, CA, 1971. + +\bibitem[Le et~al.(2017)Le, Baydin, and Wood]{le2016inference} +Tuan~Anh Le, Atılım~Güneş Baydin, and Frank Wood. +\newblock Inference compilation and universal probabilistic programming. +\newblock In \emph{Proceedings of the 20th International Conference on + Artificial Intelligence and Statistics (AISTATS)}, volume~54 of + \emph{Proceedings of Machine Learning Research}, pages 1338--1348, Fort + Lauderdale, FL, USA, 2017. PMLR. + +\bibitem[LeCun et~al.(1998)LeCun, Bottou, Bengio, and + Haffner]{lecun1998gradient} +Yann LeCun, L{\'e}on Bottou, Yoshua Bengio, and Patrick Haffner. +\newblock Gradient-based learning applied to document recognition. +\newblock \emph{Proceedings of the IEEE}, 86\penalty0 (11):\penalty0 + 2278--2324, 1998. + +\bibitem[LeCun et~al.(2015)LeCun, Bengio, and Hinton]{lecun2015deep} +Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. +\newblock Deep learning. +\newblock \emph{Nature}, 521\penalty0 (7553):\penalty0 436--444, 2015. + +\bibitem[Leibniz(1685)]{Leibniz1685} +G.~W. Leibniz. +\newblock \emph{Machina arithmetica in qua non additio tantum et subtractio sed + et multiplicatio nullo, diviso vero paene nullo animi labore peragantur}. +\newblock Hannover, 1685. + +\bibitem[Leroy(1997)]{leroy1997effectiveness} +Xavier Leroy. +\newblock The effectiveness of type-based unboxing. +\newblock In \emph{TIC 1997: Workshop Types in Compilation}, 1997. + +\bibitem[Linnainmaa(1970)]{linnainmaa1970representation} +Seppo Linnainmaa. +\newblock The representation of the cumulative rounding error of an algorithm + as a taylor expansion of the local rounding errors. +\newblock Master's thesis, University of Helsinki, 1970. + +\bibitem[Linnainmaa(1976)]{linnainmaa1976taylor} +Seppo Linnainmaa. +\newblock Taylor expansion of the accumulated rounding error. +\newblock \emph{{BIT} Numerical Mathematics}, 16\penalty0 (2):\penalty0 + 146--160, 1976. + +\bibitem[Loper and Black(2014)]{loper2014opendr} +Matthew~M. Loper and Michael~J. Black. +\newblock {OpenDR}: An approximate differentiable renderer. +\newblock In \emph{European Conference on Computer Vision}, pages 154--169. + Springer, 2014. + +\bibitem[Maclaurin(2016)]{maclaurin2016modeling} +Dougal Maclaurin. +\newblock \emph{Modeling, Inference and Optimization with Composable + Differentiable Procedures}. +\newblock PhD thesis, School of Engineering and Applied Sciences, Harvard + University, 2016. + +\bibitem[Maclaurin et~al.(2015)Maclaurin, Duvenaud, and Adams]{Maclaurin2015} +Dougal Maclaurin, David Duvenaud, and Ryan Adams. +\newblock Gradient-based hyperparameter optimization through reversible + learning. +\newblock In \emph{International Conference on Machine Learning}, pages + 2113--2122, 2015. + +\bibitem[Manzyuk et~al.(2012)Manzyuk, Pearlmutter, Radul, Rush, and + Siskind]{manzyuk2012confusion} +Oleksandr Manzyuk, Barak~A. Pearlmutter, Alexey~Andreyevich Radul, David~R + Rush, and Jeffrey~Mark Siskind. +\newblock Confusion of tagged perturbations in forward automatic + differentiation of higher-order functions. +\newblock \emph{arXiv preprint arXiv:1211.4892}, 2012. + +\bibitem[Mayne and Jacobson(1970)]{jacobson1970differential} +David~Q. Mayne and David~H. Jacobson. +\newblock \emph{Differential Dynamic Programming}. +\newblock American Elsevier Pub. Co., New York, 1970. + +\bibitem[Mazourik(1991)]{Mazourik1991} +Vladimir Mazourik. +\newblock Integration of automatic differentiation into a numerical library for + {PC}'s. +\newblock In A.~Griewank and G.~F. Corliss, editors, \emph{Automatic + Differentiation of Algorithms: Theory, Implementation, and Application}, + pages 315--29. Society for Industrial and Applied Mathematics, Philadelphia, + PA, 1991. + +\bibitem[Meyer et~al.(2003)Meyer, Fournier, and Berg]{Meyer2003} +Renate Meyer, David~A. Fournier, and Andreas Berg. +\newblock Stochastic volatility: Bayesian computation using automatic + differentiation and the extended {Kalman} filter. +\newblock \emph{Econometrics Journal}, 6\penalty0 (2):\penalty0 408--420, 2003. +\newblock \doi{10.1111/1368-423X.t01-1-00116}. + +\bibitem[Michelotti(1990)]{Michelotti1990} +L.~Michelotti. +\newblock {MXYZPTLK}: A practical, user-friendly {C++} implementation of + differential algebra: User's guide. +\newblock Technical Memorandum FN-535, Fermi National Accelerator Laboratory, + Batavia, IL, 1990. + +\bibitem[Mikolov et~al.(2010)Mikolov, Karafi{\'a}t, Burget, {\v{C}}ernock{\`y}, + and Khudanpur]{mikolov2010recurrent} +Tom{\'a}{\v{s}} Mikolov, Martin Karafi{\'a}t, Luk{\'a}{\v{s}} Burget, Jan + {\v{C}}ernock{\`y}, and Sanjeev Khudanpur. +\newblock Recurrent neural network based language model. +\newblock In \emph{Eleventh Annual Conference of the International Speech + Communication Association}, 2010. + +\bibitem[Müller and Cusdin(2005)]{Muller2005} +J.~D. Müller and P.~Cusdin. +\newblock On the performance of discrete adjoint {CFD} codes using automatic + differentiation. +\newblock \emph{International Journal for Numerical Methods in Fluids}, + 47\penalty0 (8-9):\penalty0 939--945, 2005. +\newblock ISSN 1097-0363. +\newblock \doi{10.1002/fld.885}. + +\bibitem[Naumann(2004)]{naumann2004optimal} +Uwe Naumann. +\newblock Optimal accumulation of {Jacobian} matrices by elimination methods on + the dual computational graph. +\newblock \emph{Mathematical Programming}, 99\penalty0 (3):\penalty0 399--421, + 2004. + +\bibitem[Naumann and Riehme(2005)]{Naumann2005} +Uwe Naumann and Jan Riehme. +\newblock Computing adjoints with the {NAGWare} {F}ortran~95 compiler. +\newblock In H.~M. B{\"u}cker, G.~Corliss, P.~Hovland, U.~Naumann, and + B.~Norris, editors, \emph{Automatic Differentiation: {A}pplications, Theory, + and Implementations}, Lecture Notes in Computational Science and Engineering, + pages 159--69. Springer, 2005. + +\bibitem[Neal(1993)]{Neal1993} +Radford~M. Neal. +\newblock Probabilistic inference using {Markov} chain {Monte Carlo} methods. +\newblock Technical Report CRG-TR-93-1, Department of Computer Science, + University of Toronto, 1993. + +\bibitem[Neidinger(1989)]{Neidinger1989} +Richard~D. Neidinger. +\newblock Automatic differentiation and {APL}. +\newblock \emph{College Mathematics Journal}, 20\penalty0 (3):\penalty0 + 238--51, 1989. +\newblock \doi{10.2307/2686776}. + +\bibitem[Nolan(1953)]{Nolan1953} +John~F. Nolan. +\newblock Analytical differentiation on a digital computer. +\newblock Master's thesis, Massachusetts Institute of Technology, 1953. + +\bibitem[Ostiguy and Michelotti(2007)]{Ostiguy2007} +J.~F. Ostiguy and L.~Michelotti. +\newblock Mxyzptlk: An efficient, native {C++} differentiation engine. +\newblock In \emph{Particle Accelerator Conference (PAC 2007)}, pages 3489--91. + IEEE, 2007. +\newblock \doi{10.1109/PAC.2007.4440468}. + +\bibitem[Parker(1985)]{Parker1985} +David~B. Parker. +\newblock Learning-logic: Casting the cortex of the human brain in silicon. +\newblock Technical Report TR-47, Center for Computational Research in + Economics and Management Science, MIT, 1985. + +\bibitem[Pascual and Hascoët(2008)]{Pascual2008} +Valérie Pascual and Laurent Hascoët. +\newblock {TAPENADE} for {C}. +\newblock In \emph{Advances in Automatic Differentiation}, Lecture Notes in + Computational Science and Engineering, pages 199--210. Springer, 2008. +\newblock \doi{10.1007/978-3-540-68942-3_18}. + +\bibitem[Paszke et~al.(2017)Paszke, Gross, Chintala, Chanan, Yang, DeVito, Lin, + Desmaison, Antiga, and Lerer]{paszke2017automatic} +Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary + DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. +\newblock Automatic differentiation in {PyTorch}. +\newblock In \emph{NIPS 2017 Autodiff Workshop: The Future of Gradient-based + Machine Learning Software and Techniques, Long Beach, CA, US, December 9, + 2017}, 2017. + +\bibitem[Pearlmutter(1994)]{Pearlmutter1994} +Barak~A. Pearlmutter. +\newblock Fast exact multiplication by the {Hessian}. +\newblock \emph{Neural Computation}, 6:\penalty0 147--60, 1994. +\newblock \doi{10.1162/neco.1994.6.1.147}. + +\bibitem[Pearlmutter and Siskind(2008)]{pearlmutter2008reverse} +Barak~A. Pearlmutter and Jeffrey~Mark Siskind. +\newblock Reverse-mode {AD} in a functional framework: Lambda the ultimate + backpropagator. +\newblock \emph{ACM Transactions on Programming Languages and Systems + (TOPLAS)}, 30\penalty0 (2):\penalty0 1--36, March 2008. +\newblock \doi{10.1145/1330017.1330018}. + +\bibitem[Peng and Robinson(1976)]{Peng1976} +Ding-Yu Peng and Donald~B. Robinson. +\newblock A new two-constant equation of state. +\newblock \emph{Industrial and Engineering Chemistry Fundamentals}, 15\penalty0 + (1):\penalty0 59--64, 1976. +\newblock \doi{10.1021/i160057a011}. + +\bibitem[Peterson(1989)]{peterson1989untagged} +John Peterson. +\newblock Untagged data in tagged environments: Choosing optimal + representations at compile time. +\newblock In \emph{Proceedings of the Fourth International Conference on + Functional Programming Languages and Computer Architecture}, pages 89--99. + ACM, 1989. + +\bibitem[Pfeiffer(1987)]{Pfeiffer1987} +F.~W. Pfeiffer. +\newblock Automatic differentiation in {PROSE}. +\newblock \emph{SIGNUM Newsletter}, 22\penalty0 (1):\penalty0 2--8, 1987. +\newblock \doi{10.1145/24680.24681}. + +\bibitem[Pock et~al.(2007)Pock, Pock, and Bischof]{Pock2007} +Thomas Pock, Michael Pock, and Horst Bischof. +\newblock Algorithmic differentiation: Application to variational problems in + computer vision. +\newblock \emph{IEEE Transactions on Pattern Analysis and Machine + Intelligence}, 29\penalty0 (7):\penalty0 1180--1193, 2007. +\newblock \doi{10.1109/TPAMI.2007.1044}. + +\bibitem[Press et~al.(2007)Press, Teukolsky, Vetterling, and + Flannery]{Press2007} +William~H. Press, Saul~A. Teukolsky, William~T. Vetterling, and Brian~P. + Flannery. +\newblock \emph{Numerical Recipes: The Art of Scientific Computing}. +\newblock Cambridge University Press, 2007. + +\bibitem[Rall(2006)]{Rall2006} +Louise~B. Rall. +\newblock Perspectives on automatic differentiation: Past, present, and future? +\newblock In M.~B\"{u}cker, G.~Corliss, U.~Naumann, P.~Hovland, and B.~Norris, + editors, \emph{Automatic Differentiation: Applications, Theory, and + Implementations}, volume~50 of \emph{Lecture Notes in Computational Science + and Engineering}, pages 1--14. Springer Berlin Heidelberg, 2006. + +\bibitem[Rasmussen and Williams(2006)]{Rasmussen2006} +Carl~Edward Rasmussen and Christopher K.~I. Williams. +\newblock \emph{Gaussian processes for machine learning}. +\newblock {MIT} Press, 2006. + +\bibitem[Revels et~al.(2016{\natexlab{a}})Revels, Lubin, and + Papamarkou]{RevelsLubinPapamarkou2016} +J.~Revels, M.~Lubin, and T.~Papamarkou. +\newblock Forward-mode automatic differentiation in {Julia}. +\newblock \emph{arXiv:1607.07892 [cs.MS]}, 2016{\natexlab{a}}. +\newblock URL \url{https://arxiv.org/abs/1607.07892}. + +\bibitem[Revels et~al.(2016{\natexlab{b}})Revels, Lubin, and + Papamarkou]{revels2016forward} +Jarrett Revels, Miles Lubin, and Theodore Papamarkou. +\newblock Forward-mode automatic differentiation in {J}ulia. +\newblock \emph{arXiv preprint arXiv:1607.07892}, 2016{\natexlab{b}}. + +\bibitem[Rezende et~al.(2014)Rezende, Mohamed, and + Wierstra]{rezende2014stochastic} +Danilo~Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. +\newblock Stochastic backpropagation and approximate inference in deep + generative models. +\newblock In \emph{International Conference on Machine Learning}, pages + 1278--1286, 2014. + +\bibitem[Rich and Hill(1992)]{Hill1992} +Lawrence~C. Rich and David~R. Hill. +\newblock Automatic differentiation in {MATLAB}. +\newblock \emph{Applied Numerical Mathematics}, 9:\penalty0 33--43, 1992. + +\bibitem[Ritchie et~al.(2016)Ritchie, Horsfall, and Goodman]{ritchie2016deep} +Daniel Ritchie, Paul Horsfall, and Noah~D Goodman. +\newblock Deep amortized inference for probabilistic programs. +\newblock \emph{arXiv preprint arXiv:1610.05735}, 2016. + +\bibitem[Rollins(2009)]{Rollins2009} +Elizabeth Rollins. +\newblock Optimization of neural network feedback control systems using + automatic differentiation. +\newblock Master's thesis, Department of Aeronautics and Astronautics, + Massachusetts Institute of Technology, 2009. + +\bibitem[Rozonoer(1959)]{Rozonoer-Pontryagin-1959a} +L.~I. Rozonoer. +\newblock {L}. {S}. {Pontryagin}'s maximum principle in the theory of optimum + systems---{Part} {II}. +\newblock \emph{Automat. i Telemekh.}, 20:\penalty0 1441--1458, 1959. + +\bibitem[Rumelhart et~al.(1986)Rumelhart, Hinton, and + Williams]{rumelhart1986learning} +David~E. Rumelhart, Geoffrey~E. Hinton, and Ronald~J. Williams. +\newblock Learning representations by back-propagating errors. +\newblock \emph{Nature}, 323\penalty0 (6088):\penalty0 533, 1986. + +\bibitem[Rump(1999)]{Rump1999} +Siegfried~M. Rump. +\newblock {INTLAB}---{INTerval} {LABoratory}. +\newblock In \emph{Developments in Reliable Computing}, pages 77--104. Kluwer + Academic Publishers, Dordrecht, 1999. +\newblock \doi{10.1007/978-94-017-1247-7_7}. + +\bibitem[Salimans et~al.(2015)Salimans, Kingma, and Welling]{Salimans2014} +Tim Salimans, Diederik Kingma, and Max Welling. +\newblock Markov chain {Monte Carlo} and variational inference: Bridging the + gap. +\newblock In \emph{Proceedings of the 32nd International Conference on Machine + Learning (ICML-15)}, pages 1218--1226, 2015. + +\bibitem[Salvatier et~al.(2016)Salvatier, Wiecki, and + Fonnesbeck]{salvatier2016probabilistic} +John Salvatier, Thomas~V Wiecki, and Christopher Fonnesbeck. +\newblock Probabilistic programming in {Python} using {PyMC3}. +\newblock \emph{PeerJ Computer Science}, 2:\penalty0 e55, 2016. + +\bibitem[Schaul et~al.(2013)Schaul, Zhang, and LeCun]{schaul2013no} +Tom Schaul, Sixin Zhang, and Yann LeCun. +\newblock No more pesky learning rates. +\newblock In \emph{International Conference on Machine Learning}, pages + 343--351, 2013. + +\bibitem[Schmidhuber(2015)]{schmidhuber2015deep} +J{\"u}rgen Schmidhuber. +\newblock Deep learning in neural networks: An overview. +\newblock \emph{Neural Networks}, 61:\penalty0 85--117, 2015. + +\bibitem[Schraudolph(1999)]{Schraudolph1999} +Nicol~N. Schraudolph. +\newblock Local gain adaptation in stochastic gradient descent. +\newblock In \emph{Proceedings of the International Conference on Artificial + Neural Networks}, pages 569--74, Edinburgh, Scotland, 1999. IEE London. +\newblock \doi{10.1049/cp:19991170}. + +\bibitem[Schraudolph and Graepel(2003)]{Schraudolph2003} +Nicol~N. Schraudolph and Thore Graepel. +\newblock Combining conjugate direction methods with stochastic approximation + of gradients. +\newblock In \emph{Proceedings of the Ninth International Workshop on + Artificial Intelligence and Statistics}, 2003. + +\bibitem[Seide and Agarwal(2016)]{seide2016cntk} +Frank Seide and Amit Agarwal. +\newblock {CNTK}: Microsoft's open-source deep-learning toolkit. +\newblock In \emph{Proceedings of the 22Nd ACM SIGKDD International Conference + on Knowledge Discovery and Data Mining}, KDD '16, pages 2135--2135, New York, + NY, USA, 2016. ACM. +\newblock ISBN 978-1-4503-4232-2. +\newblock \doi{10.1145/2939672.2945397}. + +\bibitem[Shazeer et~al.(2017)Shazeer, Mirhoseini, Maziarz, Davis, Le, Hinton, + and Dean]{shazeer2017outrageously} +Noam Shazeer, Azalia Mirhoseini, Krzysztof Maziarz, Andy Davis, Quoc Le, + Geoffrey Hinton, and Jeff Dean. +\newblock Outrageously large neural networks: The sparsely-gated + mixture-of-experts layer. +\newblock In \emph{International Conference on Learning Representations 2017}, + 2017. + +\bibitem[Shivers(1991)]{shivers1991control} +Olin Shivers. +\newblock \emph{Control-flow analysis of higher-order languages}. +\newblock PhD thesis, Carnegie Mellon University, 1991. + +\bibitem[Shtof et~al.(2013)Shtof, Agathos, Gingold, Shamir, and + {Cohen‐Or}]{Shtof2013} +Alex Shtof, Alexander Agathos, Yotam Gingold, Ariel Shamir, and Daniel + {Cohen‐Or}. +\newblock Geosemantic snapping for sketch-based modeling. +\newblock \emph{Computer Graphics Forum}, 32\penalty0 (2):\penalty0 245--53, + 2013. +\newblock \doi{10.1111/cgf.12044}. + +\bibitem[Siddharth et~al.(2017)Siddharth, Paige, van~de Meent, Desmaison, + Goodman, Kohli, Wood, and Torr]{siddharth2017learning} +N.~Siddharth, Brooks Paige, Jan-Willem van~de Meent, Alban Desmaison, Noah~D. + Goodman, Pushmeet Kohli, Frank Wood, and Philip Torr. +\newblock Learning disentangled representations with semi-supervised deep + generative models. +\newblock In I.~Guyon, U.~V. Luxburg, S.~Bengio, H.~Wallach, R.~Fergus, + S.~Vishwanathan, and R.~Garnett, editors, \emph{Advances in Neural + Information Processing Systems 30}, pages 5927--5937. Curran Associates, + Inc., 2017. + +\bibitem[Simard et~al.(1998)Simard, {LeCun}, Denker, and Victorri]{Simard1998} +Patrice Simard, Yann {LeCun}, John Denker, and Bernard Victorri. +\newblock Transformation invariance in pattern recognition, tangent distance + and tangent propagation. +\newblock In G.~Orr and K.~Muller, editors, \emph{Neural Networks: Tricks of + the Trade}. Springer, 1998. + +\bibitem[Sirkes and Tziperman(1997)]{sirkes-tziperman-1997a} +Z.~Sirkes and E.~Tziperman. +\newblock Finite difference of adjoint or adjoint of finite difference? +\newblock \emph{Monthly Weather Review}, 125\penalty0 (12):\penalty0 3373--8, + 1997. +\newblock \doi{10.1175/1520-0493(1997)125<3373:FDOAOA>2.0.CO;2}. + +\bibitem[Siskind and Pearlmutter(2005)]{SiskindPearlmutter2005a} +Jeffrey~Mark Siskind and Barak~A. Pearlmutter. +\newblock Perturbation confusion and referential transparency: Correct + functional implementation of forward-mode {AD}. +\newblock In Andrew Butterfield, editor, \emph{Implementation and Application + of Functional Languages---17th International Workshop, IFL'05}, pages 1--9, + Dublin, Ireland, 2005. +\newblock Trinity College Dublin Computer Science Department Technical Report + TCD-CS-2005-60. + +\bibitem[Siskind and Pearlmutter(2008{\natexlab{a}})]{Siskind2008} +Jeffrey~Mark Siskind and Barak~A. Pearlmutter. +\newblock Using polyvariant union-free flow analysis to compile a higher-order + functional-programming language with a first-class derivative operator to + efficient {Fortran}-like code. +\newblock Technical Report TR-ECE-08-01, School of Electrical and Computer + Engineering, Purdue University, 2008{\natexlab{a}}. + +\bibitem[Siskind and Pearlmutter(2008{\natexlab{b}})]{Siskind2008b} +Jeffrey~Mark Siskind and Barak~A. Pearlmutter. +\newblock Nesting forward-mode {AD} in a functional framework. +\newblock \emph{Higher-Order and Symbolic Computation}, 21\penalty0 + (4):\penalty0 361--376, 2008{\natexlab{b}}. + +\bibitem[Siskind and Pearlmutter(2016)]{siskind2016efficient} +Jeffrey~Mark Siskind and Barak~A. Pearlmutter. +\newblock Efficient implementation of a higher-order language with built-in + {AD}. +\newblock In \emph{7th International Conference on Algorithmic Differentiation, + Christ Church Oxford, UK, September 12--15, 2016}, 2016. +\newblock Also arXiv:1611.03416. + +\bibitem[Siskind and Pearlmutter(2017)]{siskind2017divide} +Jeffrey~Mark Siskind and Barak~A. Pearlmutter. +\newblock Divide-and-conquer checkpointing for arbitrary programs with no user + annotation. +\newblock In \emph{NIPS 2017 Autodiff Workshop: The Future of Gradient-based + Machine Learning Software and Techniques, Long Beach, CA, US, December 9, + 2017}, 2017. +\newblock Also arXiv:1708.06799. + +\bibitem[Slusanschi and Dumitrel(2016)]{slusanschi2016adijac} +Emil~I. Slusanschi and Vlad Dumitrel. +\newblock {ADiJaC}---{A}utomatic differentiation of {J}ava classfiles. +\newblock \emph{ACM Transaction on Mathematical Software}, 43\penalty0 + (2):\penalty0 9:1--9:33, September 2016. +\newblock ISSN 0098-3500. +\newblock \doi{10.1145/2904901}. + +\bibitem[Speelpenning(1980)]{Speelpenning80} +Bert Speelpenning. +\newblock \emph{Compiling Fast Partial Derivatives of Functions Given by + Algorithms}. +\newblock PhD thesis, Department of Computer Science, University of Illinois at + Urbana-Champaign, 1980. + +\bibitem[Sra et~al.(2011)Sra, Nowozin, and Wright]{Sra2011} +Suvrit Sra, Sebastian Nowozin, and Stephen~J. Wright. +\newblock \emph{Optimization for Machine Learning}. +\newblock {MIT} Press, 2011. + +\bibitem[Srajer et~al.(2016)Srajer, Kukelova, and + Fitzgibbon]{srajer2016benchmark} +Filip Srajer, Zuzana Kukelova, and Andrew Fitzgibbon. +\newblock A benchmark of selected algorithmic differentiation tools on some + problems in machine learning and computer vision. +\newblock In \emph{AD2016: The 7th International Conference on Algorithmic + Differentiation, Monday 12th--Thursday 15th September 2016, Christ Church + Oxford, UK: Programme and Abstracts}, pages 181--184. Society for Industrial + and Applied Mathematics (SIAM), 2016. + +\bibitem[Srinivasan and Todorov(2015)]{srinivasan-todorov-2015a} +Akshay Srinivasan and Emanuel Todorov. +\newblock Graphical {Newton}. +\newblock Technical Report arXiv:1508.00952, arXiv preprint, 2015. + +\bibitem[Stuhlm{\"u}ller et~al.(2013)Stuhlm{\"u}ller, Taylor, and + Goodman]{stuhlmuller2013learning} +Andreas Stuhlm{\"u}ller, Jacob Taylor, and Noah Goodman. +\newblock Learning stochastic inverses. +\newblock In \emph{Advances in Neural Information Processing Systems}, pages + 3048--3056, 2013. + +\bibitem[Such et~al.(2017)Such, Madhavan, Conti, Lehman, Stanley, and + Clune]{such2017deep} +Felipe~Petroski Such, Vashisht Madhavan, Edoardo Conti, Joel Lehman, Kenneth~O. + Stanley, and Jeff Clune. +\newblock Deep neuroevolution: Genetic algorithms are a competitive alternative + for training deep neural networks for reinforcement learning. +\newblock \emph{arXiv preprint arXiv:1712.06567}, 2017. + +\bibitem[Sukhbaatar et~al.(2015)Sukhbaatar, Weston, Fergus, + et~al.]{sukhbaatar2015end} +Sainbayar Sukhbaatar, Jason Weston, Rob Fergus, et~al. +\newblock End-to-end memory networks. +\newblock In \emph{Advances in Neural Information Processing Systems}, pages + 2440--2448, 2015. + +\bibitem[Sussman and Wisdom(2001)]{Sussman2001} +Gerald~J. Sussman and Jack Wisdom. +\newblock \emph{Structure and Interpretation of Classical Mechanics}. +\newblock {MIT} Press, 2001. +\newblock \doi{10.1063/1.1457268}. + +\bibitem[Taylor et~al.(2014)Taylor, Stebbing, Ramakrishna, Keskin, Shotton, + Izadi, Hertzmann, and Fitzgibbon]{taylor2014user} +Jonathan Taylor, Richard Stebbing, Varun Ramakrishna, Cem Keskin, Jamie + Shotton, Shahram Izadi, Aaron Hertzmann, and Andrew Fitzgibbon. +\newblock User-specific hand modeling from monocular depth sequences. +\newblock In \emph{Proceedings of the {IEEE} Conference on Computer Vision and + Pattern Recognition}, pages 644--651, 2014. + +\bibitem[Thomas et~al.(2006)Thomas, Dowell, and Hall]{thomas2006using} +Jeffrey~P. Thomas, Earl~H. Dowell, and Kenneth~C. Hall. +\newblock Using automatic differentiation to create a nonlinear reduced order + model of a computational fluid dynamic solver. +\newblock \emph{AIAA Paper}, 7115:\penalty0 2006, 2006. + +\bibitem[Tieleman and Hinton(2012)]{tieleman2012lecture} +T.~Tieleman and G.~Hinton. +\newblock Lecture 6.5---{RMSProp}: Divide the gradient by a running average of + its recent magnitude. +\newblock \emph{COURSERA: Neural Networks for Machine Learning}, 4\penalty0 + (2), 2012. + +\bibitem[Tokui et~al.(2015)Tokui, Oono, Hido, and Clayton]{tokui2015chainer} +Seiya Tokui, Kenta Oono, Shohei Hido, and Justin Clayton. +\newblock Chainer: a next-generation open source framework for deep learning. +\newblock In \emph{Proceedings of Workshop on Machine Learning Systems + (LearningSys) in The Twenty-ninth Annual Conference on Neural Information + Processing Systems (NIPS)}, 2015. + +\bibitem[Tran et~al.(2016)Tran, Kucukelbir, Dieng, Rudolph, Liang, and + Blei]{tran2016edward} +Dustin Tran, Alp Kucukelbir, Adji~B. Dieng, Maja Rudolph, Dawen Liang, and + David~M. Blei. +\newblock {Edward: A library for probabilistic modeling, inference, and + criticism}. +\newblock \emph{arXiv preprint arXiv:1610.09787}, 2016. + +\bibitem[Tran et~al.(2017)Tran, Hoffman, Saurous, Brevdo, Murphy, and + Blei]{tran2017deep} +Dustin Tran, Matthew~D. Hoffman, Rif~A. Saurous, Eugene Brevdo, Kevin Murphy, + and David~M. Blei. +\newblock Deep probabilistic programming. +\newblock In \emph{International Conference on Learning Representations}, 2017. + +\bibitem[Triggs et~al.(1999)Triggs, McLauchlan, Hartley, and + Fitzgibbon]{triggs1999bundle} +Bill Triggs, Philip~F. McLauchlan, Richard~I. Hartley, and Andrew~W. + Fitzgibbon. +\newblock Bundle adjustment—a modern synthesis. +\newblock In \emph{International Workshop on Vision Algorithms}, pages + 298--372. Springer, 1999. + +\bibitem[Tucker et~al.(2017)Tucker, Mnih, Maddison, Lawson, and + Sohl-Dickstein]{tucker2017rebar} +George Tucker, Andriy Mnih, Chris~J. Maddison, John Lawson, and Jascha + Sohl-Dickstein. +\newblock {REBAR}: Low-variance, unbiased gradient estimates for discrete + latent variable models. +\newblock In \emph{Advances in Neural Information Processing Systems}, pages + 2624--2633, 2017. + +\bibitem[{van Merri{\"e}nboer} et~al.(2017){van Merri{\"e}nboer}, Wiltschko, + and Moldovan]{van2017tangent} +Bart {van Merri{\"e}nboer}, Alexander~B. Wiltschko, and Dan Moldovan. +\newblock Tangent: Automatic differentiation using source code transformation + in {Python}. +\newblock \emph{arXiv preprint arXiv:1711.02712}, 2017. + +\bibitem[Verma(2000)]{Verma2000} +Arun Verma. +\newblock An introduction to automatic differentiation. +\newblock \emph{Current Science}, 78\penalty0 (7):\penalty0 804--7, 2000. + +\bibitem[Vishwanathan et~al.(2006)Vishwanathan, Schraudolph, Schmidt, and + Murphy]{Vishwanathan2006} +S.~V.~N. Vishwanathan, Nicol~N. Schraudolph, Mark~W. Schmidt, and Kevin~P. + Murphy. +\newblock Accelerated training of conditional random fields with stochastic + gradient methods. +\newblock In \emph{Proceedings of the 23rd International Conference on Machine + Learning (ICML '06)}, pages 969--76, 2006. +\newblock \doi{10.1145/1143844.1143966}. + +\bibitem[Walther(2007)]{Walther2007} +Andrea Walther. +\newblock Automatic differentiation of explicit {Runge}-{Kutta} methods for + optimal control. +\newblock \emph{Computational Optimization and Applications}, 36\penalty0 + (1):\penalty0 83--108, 2007. +\newblock \doi{10.1007/s10589-006-0397-3}. + +\bibitem[Walther and Griewank(2012)]{Walther2012} +Andrea Walther and Andreas Griewank. +\newblock Getting started with {ADOL-C}. +\newblock In U.~Naumann and O.~Schenk, editors, \emph{Combinatorial Scientific + Computing}, chapter~7, pages 181--202. Chapman-Hall CRC Computational + Science, 2012. +\newblock \doi{10.1201/b11644-8}. + +\bibitem[Wengert(1964)]{Wengert1964} +Robert~E. Wengert. +\newblock A simple automatic derivative evaluation program. +\newblock \emph{Communications of the {ACM}}, 7:\penalty0 463--4, 1964. + +\bibitem[Werbos(1974)]{Werbos-1974a} +Paul~J. Werbos. +\newblock \emph{Beyond Regression: New Tools for Prediction and Analysis in the + Behavioral Sciences}. +\newblock PhD thesis, Harvard University, 1974. + +\bibitem[Williams(1992)]{williams1992simple} +Ronald~J. Williams. +\newblock Simple statistical gradient-following algorithms for connectionist + reinforcement learning. +\newblock \emph{Machine Learning}, 8\penalty0 (3-4):\penalty0 229--256, 1992. + +\bibitem[Willkomm and Vehreschild(2013)]{Willkomm2013} +J.~Willkomm and A.~Vehreschild. +\newblock The {ADiMat} handbook, 2013. +\newblock URL \url{http://adimat.sc.informatik.tu-darmstadt.de/doc/}. + +\bibitem[Wingate et~al.(2011)Wingate, Goodman, Stuhlmüller, and + Siskind]{Wingate2011} +David Wingate, Noah Goodman, Andreas Stuhlmüller, and Jeffrey~Mark Siskind. +\newblock Nonstandard interpretations of probabilistic programs for efficient + inference. +\newblock \emph{Advances in Neural Information Processing Systems}, 23, 2011. + +\bibitem[Yang et~al.(2008)Yang, Zhao, Yan, and Chen]{Yang2008} +Weiwei Yang, Yong Zhao, Li~Yan, and Xiaoqian Chen. +\newblock Application of {PID} controller based on {BP} neural network using + automatic differentiation method. +\newblock In F.~Sun, J.~Zhang, Y.~Tan, J.~Cao, and W.~Yu, editors, + \emph{Advances in Neural Networks---ISNN 2008}, volume 5264 of \emph{Lecture + Notes in Computer Science}, pages 702--711. Springer Berlin Heidelberg, 2008. +\newblock \doi{10.1007/978-3-540-87734-9_80}. + +\bibitem[Yildirim et~al.(2015)Yildirim, Kulkarni, Freiwald, and + Tenenbaum]{yildirim2015efficient} +Ilker Yildirim, Tejas~D. Kulkarni, Winrich~A. Freiwald, and Joshua~B. + Tenenbaum. +\newblock Efficient and robust analysis-by-synthesis in vision: A computational + framework, behavioral tests, and modeling neuronal representations. +\newblock In \emph{Annual Conference of the Cognitive Science Society}, 2015. + +\bibitem[Yu and Siskind(2013)]{Yu2013} +Haonan Yu and Jeffrey~Mark Siskind. +\newblock Grounded language learning from video described with sentences. +\newblock In \emph{Proceedings of the 51st Annual Meeting of the Association + for Computational Linguistics}, pages 53--63, Sofia, Bulgaria, 2013. + Association for Computational Linguistics. + +\bibitem[Zaremba et~al.(2016)Zaremba, Mikolov, Joulin, and + Fergus]{zaremba2016learning} +Wojciech Zaremba, Tomas Mikolov, Armand Joulin, and Rob Fergus. +\newblock Learning simple algorithms from examples. +\newblock In \emph{International Conference on Machine Learning}, pages + 421--429, 2016. + +\bibitem[Zhu et~al.(1997)Zhu, Byrd, Lu, and Nocedal]{Zhu1997} +Ciyou Zhu, Richard~H. Byrd, Peihuang Lu, and Jorge Nocedal. +\newblock Algorithm 778: {L-BFGS-B}: {Fortran} subroutines for large-scale + bound-constrained optimization. +\newblock \emph{ACM Transactions on Mathematical Software (TOMS)}, 23\penalty0 + (4):\penalty0 550--60, 1997. +\newblock \doi{10.1145/279232.279236}. + +\end{thebibliography} diff --git a/doc/Articles/Autodiff/17-468.tex b/doc/Articles/Autodiff/17-468.tex new file mode 100644 index 000000000..6a62db0d2 --- /dev/null +++ b/doc/Articles/Autodiff/17-468.tex @@ -0,0 +1,694 @@ +\documentclass[twoside,11pt]{article} + +% Any additional packages needed should be included after jmlr2e. +% Note that jmlr2e.sty includes epsfig, amssymb, natbib and graphicx, +% and defines many common macros, such as 'proof' and 'example'. +% +% It also sets the bibliographystyle to plainnat; for more information on +% natbib citation styles, see the natbib documentation, a copy of which +% is archived at http://www.jmlr.org/format/natbib.pdf + +\PassOptionsToPackage{hyphens}{url} +\usepackage{jmlr2e_mod} +\usepackage[utf8]{inputenc} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage[inline]{enumitem} +\usepackage{paralist} +\usepackage{adjustbox} +\usepackage{tikz} +\usetikzlibrary{decorations.pathmorphing} +\usetikzlibrary{arrows} +\usetikzlibrary{positioning} +\usepackage{standalone} +\usepackage{rotating} +\usepackage{tabularx} +\usepackage{booktabs} +\usepackage{multirow} +\usepackage{todonotes} +\usepackage{microtype} +\hyphenation{DiffSharp} +\hyphenation{TensorFlow} +\hyphenation{auto-diff} +\hyphenation{ADIFOR} + +% Definitions of handy macros can go here +\newcommand{\dataset}{{\cal D}} +\newcommand{\fracpartial}[2]{\frac{\partial #1}{\partial #2}} +\DeclareMathOperator*{\argmin}{arg\,min} +\newcommand{\Del}{\mathrm{\Delta}} +\newcommand{\RedMark}{{\color[rgb]{1,0,0}$X$}} +\newcommand{\GreenMark}{{\color[rgb]{0,0.69,0}$Y$}} +\newcommand{\BlueMark}{{\color[rgb]{0,0,1}$Z$}} + +% Heading arguments are {volume}{year}{pages}{date submitted}{date published}{paper id}{author-full-names} +% \jmlrheading{1}{2000}{1-48}{4/00}{10/00}{meila00a}{Marina Meil\u{a} and Michael I. Jordan} + +\jmlrheading{18}{2018}{1-43}{8/17}{2/18}{17-468}{Atılım Güneş Baydin, Barak~A.~Pearlmutter, Alexey~Andreyevich~Radul, and Jeffrey~Mark~Siskind} + +% Short headings should be running head and authors last names + +\ShortHeadings{Automatic Differentiation in Machine Learning: a Survey}{Baydin, Pearlmutter, Radul, and Siskind} +\firstpageno{1} + +\begin{document} + +\title{Automatic Differentiation\\in Machine Learning: a Survey} + +\author{\name Atılım Güneş Baydin \email gunes@robots.ox.ac.uk \\ + \addr Department of Engineering Science\\ + University of Oxford\\ + Oxford OX1 3PJ, United Kingdom + \AND + \name Barak~A.~Pearlmutter \email barak@pearlmutter.net \\ + \addr Department of Computer Science\\ + National University of Ireland Maynooth\\ + Maynooth, Co. Kildare, Ireland + \AND + \name Alexey~Andreyevich~Radul \email axch@mit.edu \\ + \addr Department of Brain and Cognitive Sciences\\ + Massachusetts Institute of Technology\\ + Cambridge, MA 02139, United States + \AND + \name Jeffrey~Mark~Siskind \email qobi@purdue.edu \\ + \addr School of Electrical and Computer Engineering\\ + Purdue University\\ + West Lafayette, IN 47907, United States} + +\editor{Léon Bottou} + +\maketitle + +\begin{abstract}% <- trailing '%' for backward compatibility of .sty file +Derivatives, mostly in the form of gradients and Hessians, are ubiquitous in machine learning. Automatic differentiation (AD), also called algorithmic differentiation or simply ``autodiff'', is a family of techniques similar to but more general than backpropagation for efficiently and accurately evaluating derivatives of numeric functions expressed as computer programs. AD is a small but established field with applications in areas including computational fluid dynamics, atmospheric sciences, and engineering design optimization. Until very recently, the fields of machine learning and AD have largely been unaware of each other and, in some cases, have independently discovered each other's results. Despite its relevance, general-purpose AD has been missing from the machine learning toolbox, a situation slowly changing with its ongoing adoption under the names ``dynamic computational graphs'' and ``differentiable programming''. We survey the intersection of AD and machine learning, cover applications where AD has direct relevance, and address the main implementation techniques. By precisely defining the main differentiation techniques and their interrelationships, we aim to bring clarity to the usage of the terms ``autodiff'', ``automatic differentiation'', and ``symbolic differentiation'' as these are encountered more and more in machine learning settings. + +\end{abstract} + +\begin{keywords} + Backpropagation, Differentiable Programming +\end{keywords} + +\section{Introduction} + +Methods for the computation of derivatives in computer programs can be classified into four categories: +\begin{inparaenum}[(1)] + \item manually working out derivatives and coding them; + \item \emph{numerical differentiation} using finite difference approximations; + \item \emph{symbolic differentiation} using expression manipulation in computer algebra systems such as Mathematica, Maxima, and Maple; and + \item \emph{automatic differentiation}, also called \emph{algorithmic differentiation}, which is the subject matter of this paper. +\end{inparaenum} + +Conventionally, many methods in machine learning have required the evaluation of derivatives and most of the traditional learning algorithms have relied on the computation of gradients and Hessians of an objective function \citep{Sra2011}. When introducing new models, machine learning researchers have spent considerable effort on the manual derivation of analytical derivatives to subsequently plug these into standard optimization procedures such as L-BFGS \citep{Zhu1997} or stochastic gradient descent \citep{Bottou1998}. Manual differentiation is time consuming and prone to error. Of the other alternatives, numerical differentiation is simple to implement but can be highly inaccurate due to round-off and truncation errors \citep{Jerrell1997}; more importantly, it scales poorly for gradients, rendering it inappropriate for machine learning where gradients with respect to millions of parameters are commonly needed. Symbolic differentiation addresses the weaknesses of both the manual and numerical methods, but often results in complex and cryptic expressions plagued with the problem of ``expression swell'' \citep{Corliss1988}. Furthermore, manual and symbolic methods require models to be defined as closed-form expressions, ruling out or severely limiting algorithmic control flow and expressivity. + +We are concerned with the powerful fourth technique, automatic differentiation (AD). AD performs a non-standard interpretation of a given computer program by replacing the domain of the variables to incorporate derivative values and redefining the semantics of the operators to propagate derivatives per the chain rule of differential calculus. Despite its widespread use in other fields, general-purpose AD has been underused by the machine learning community until very recently.\footnote{See, e.g., \url{https://justindomke.wordpress.com/2009/02/17/automatic-differentiation-the-most-criminally-underused-tool-in-the-potential-machine-learning-toolbox/}} Following the emergence of deep learning \citep{lecun2015deep,goodfellow2016deep} as the state-of-the-art in many machine learning tasks and the modern workflow based on rapid prototyping and code reuse in frameworks such as Theano \citep{Bastien2012}, Torch \citep{collobert2011torch7}, and TensorFlow \citep{abadi2016tensorflow}, the situation is slowly changing where projects such as autograd\footnote{\url{https://github.com/HIPS/autograd}} \citep{maclaurin2016modeling}, Chainer\footnote{\url{https://chainer.org/}} \citep{tokui2015chainer}, and PyTorch\footnote{\url{http://pytorch.org/}} \citep{paszke2017automatic} are leading the way in bringing general-purpose AD to the mainstream. + +The term ``automatic'' in AD can be a source of confusion, causing machine learning practitioners to put the label ``automatic differentiation'', or just ``autodiff'', on any method or tool that does not involve manual differentiation, without giving due attention to the underlying mechanism. We would like to stress that AD as a technical term refers to a specific family of techniques that compute derivatives through accumulation of values during code execution to generate numerical derivative evaluations rather than derivative expressions. This allows accurate evaluation of derivatives at machine precision with only a small constant factor of overhead and ideal asymptotic efficiency. In contrast with the effort involved in arranging code as closed-form expressions under the syntactic and semantic constraints of symbolic differentiation, AD can be applied to regular code with minimal change, allowing branching, loops, and recursion. Because of this generality, AD has been applied to computer simulations in industry and academia and found applications in fields including engineering design optimization \citep{forth2002aerofoil,casanova2002application}, computational fluid dynamics \citep{Muller2005,thomas2006using,Bischof2006}, physical modeling \citep{Ekstrom2010}, optimal control \citep{Walther2007}, structural mechanics \citep{haase2002optimal}, atmospheric sciences \citep{Carmichael1997,Charpentier2000}, and computational finance \citep{Bischof2002,Capriotti2011}. + +In machine learning, a specialized counterpart of AD known as the backpropagation algorithm has been the mainstay for training neural networks, with a colorful history of having been reinvented at various times by independent researchers \citep{Griewank2012,schmidhuber2015deep}. It has been one of the most studied and used training algorithms since the day it became popular mainly through the work of \citet{rumelhart1986learning}. In simplest terms, backpropagation models learning as gradient descent in neural network weight space, looking for the minima of an objective function. The required gradient is obtained by the backward propagation of the sensitivity of the objective value at the output (Figure~\ref{FigureBackpropagation}), utilizing the chain rule to compute partial derivatives of the objective with respect to each weight. The resulting algorithm is essentially equivalent to transforming the network evaluation function composed with the objective function under reverse mode AD, which, as we shall see, actually generalizes the backpropagation idea. Thus, a modest understanding of the mathematics underlying backpropagation provides one with sufficient background for grasping AD techniques. + +\begin{figure} + \centering + \trimbox{0cm -0.4cm}{\resizebox{0.75\textwidth}{!}{\includegraphics{figures/backprop/backprop}}} + \caption{Overview of backpropagation. (a) Training inputs $x_i$ are fed forward, generating corresponding activations $y_i$. An error $E$ between the actual output $y_3$ and the target output $t$ is computed. (b) The error adjoint is propagated backward, giving the gradient with respect to the weights $\nabla_{w_i}E = \left(\frac{\partial E}{\partial w_1},\dots,\frac{\partial E}{\partial w_6}\right)$, which is subsequently used in a gradient-descent procedure. The gradient with respect to inputs $\nabla_{x_i}E$ can be also computed in the same backward pass.} + \label{FigureBackpropagation} +\end{figure} + +In this paper we review AD from a machine learning perspective, covering its origins, applications in machine learning, and methods of implementation. Along the way, we also aim to dispel some misconceptions that we believe have impeded wider recognition of AD by the machine learning community. In Section~\ref{SectionWhatADIsNot} we start by explicating how AD differs from numerical and symbolic differentiation. Section~\ref{SectionPreliminaries} gives an introduction to the AD technique and its forward and reverse accumulation modes. Section~\ref{SectionDerivativesAndMachineLearning} discusses the role of derivatives in machine learning and examines cases where AD has relevance. Section~\ref{SectionImplementations} covers various implementation approaches and general-purpose AD tools, followed by Section~\ref{SectionConclusions} where we discuss future directions. + +\section{What AD Is Not} +\label{SectionWhatADIsNot} + +Without proper introduction, one might assume that AD is either a type of numerical or symbolic differentiation. Confusion can arise because AD does in fact provide numerical values of derivatives (as opposed to derivative expressions) and it does so by using symbolic rules of differentiation (but keeping track of derivative values as opposed to the resulting expressions), giving it a two-sided nature that is partly symbolic and partly numerical \citep{Griewank2003}. We start by emphasizing how AD is different from, and in several aspects superior to, these two commonly encountered techniques of computing derivatives. + +\begin{figure*} + \centering + \trimbox{0cm -0.4cm}{\resizebox{\textwidth}{!}{\small\input{figures/differentiation/differentiation.tex}}} + \caption{The range of approaches for differentiating mathematical expressions and computer code, looking at the example of a truncated logistic map (upper left). Symbolic differentiation (center right) gives exact results but requires closed-form input and suffers from expression swell; numerical differentiation (lower right) has problems of accuracy due to round-off and truncation errors; automatic differentiation (lower left) is as accurate as symbolic differentiation with only a constant factor of overhead and support for control flow.} + \label{FigureDifferentiation} +\end{figure*} + +\subsection{AD Is Not Numerical Differentiation} + +Numerical differentiation is the finite difference approximation of derivatives using values of the original function evaluated at some sample points \citep{Burden2001} (Figure~\ref{FigureDifferentiation}, lower right). In its simplest form, it is based on the limit definition of a derivative. For example, for a multivariate function $f:\mathbb{R}^n \to \mathbb{R}$, one can approximate the gradient $\nabla f=\left(\frac{\partial f}{\partial x_1},\dots,\frac{\partial f}{\partial x_n}\right)$ using +\begin{equation} + \label{EquationForwardDifference} + \frac{\partial f(\mathbf{x})}{\partial x_i} \approx \frac{f(\mathbf{x} + h \mathbf{e}_i) - f(\mathbf{x})}{h}\;, +\end{equation} +where $\mathbf{e}_i$ is the $i$-th unit vector and $h > 0$ is a small step size. This has the advantage of being uncomplicated to implement, but the disadvantages of performing $O(n)$ evaluations of $f$ for a gradient in $n$ dimensions and requiring careful consideration in selecting the step size $h$. + +Numerical approximations of derivatives are inherently ill-conditioned and unstable,\footnote{Using the limit definition of the derivative for finite difference approximation commits both cardinal sins of numerical analysis: \emph{``thou shalt not add small numbers to big numbers''}, and \emph{``thou shalt not subtract numbers which are approximately equal''}.} with the exception of complex variable methods that are applicable to a limited set of holomorphic functions \citep{Fornberg1981}. This is due to the introduction of truncation\footnote{Truncation error is the error of approximation, or inaccuracy, one gets from $h$ not actually being zero. It is proportional to a power of $h$.} and round-off\footnote{Round-off error is the inaccuracy one gets from valuable low-order bits of the final answer having to compete for machine-word space with high-order bits of $f(\mathbf{x} + h \mathbf{e}_i)$ and $f(\mathbf{x})$ (Eq.~\ref{EquationForwardDifference}), which the computer has to store just until they cancel in the subtraction at the end. Round-off error is inversely proportional to a power of $h$.} errors inflicted by the limited precision of computations and the chosen value of the step size $h$. Truncation error tends to zero as $h \to 0$. However, as $h$ is decreased, round-off error increases and becomes dominant (Figure~\ref{FigureApproximationError}). + +\begin{figure*} + \centering + \resizebox{0.82\textwidth}{!}{\small\input{figures/approx-error/approx-error.tex}} + \caption{Error in the forward (Eq.~\ref{EquationForwardDifference}) and center difference (Eq.~\ref{EquationCenterDifference}) approximations as a function of step size $h$, for the derivative of the truncated logistic map ${f(x)=64x(1 - x)(1 - 2 x)^2 (1 - 8 x + 8 x^2)^2}$. Plotted errors are computed using ${E_{\mathrm{forward}}(h,x_0)=\left|\frac{f(x_0+h)-f(x_0)}{h} - \frac{d}{dx}f(x)\big|_{x_0}\right|}$ and ${E_{\mathrm{center}}(h,x_0)=\left|\frac{f(x_0+h)-f(x_0-h)}{2h} - \frac{d}{dx}f(x)\big|_{x_0}\right|}$ at $x_0=0.2$\;.} + \label{FigureApproximationError} +\end{figure*} + +Various techniques have been developed to mitigate approximation errors in numerical differentiation, such as using a center difference approximation +\begin{equation} + \label{EquationCenterDifference} + \frac{\partial f(\mathbf{x})}{\partial x_i} = \frac{f(\mathbf{x} + h \mathbf{e}_i) - f(\mathbf{x} - h \mathbf{e}_i)}{2h} + O(h^{2})\;, +\end{equation} +where the first-order errors cancel and one effectively moves the truncation error from first-order to second-order in $h$.\footnote{This does not avoid either of the cardinal sins, and is still highly inaccurate due to truncation.} For the one-dimensional case, it is just as costly to compute the forward difference (Eq.~\ref{EquationForwardDifference}) and the center difference (Eq.~\ref{EquationCenterDifference}), requiring only two evaluations of $f$. However, with increasing dimensionality, a trade-off between accuracy and performance is faced, where computing a Jacobian matrix of a function $f: \mathbb{R}^n \to \mathbb{R}^m$ requires $2mn$ evaluations. + +Other techniques for improving numerical differentiation, including higher-order finite differences, Richardson extrapolation to the limit \citep{Brezinski1991}, and differential quadrature methods using weighted sums \citep{Bert1996}, have increased computational complexity, do not completely eliminate approximation errors, and remain highly susceptible to floating point truncation. + +The $O(n)$ complexity of numerical differentiation for a gradient in $n$ dimensions is the main obstacle to its usefulness in machine learning, where $n$ can be as large as millions or billions in state-of-the-art deep learning models \citep{shazeer2017outrageously}. In contrast, approximation errors would be tolerated in a deep learning setting thanks to the well-documented error resiliency of neural network architectures \citep{gupta2015deep}. + +\subsection{AD Is Not Symbolic Differentiation} + +Symbolic differentiation is the automatic manipulation of expressions for obtaining derivative expressions \citep{Grabmeier2003} (Figure~\ref{FigureDifferentiation}, center right), carried out by applying transformations representing rules of differentiation such as +\begin{equation} +\begin{aligned} +\frac{d}{dx} \left(f(x) + g(x)\right) &\leadsto \frac{d}{dx} f(x) + \frac{d}{dx} g(x)\\ +\frac{d}{dx} \left(f(x)\,g(x)\right) &\leadsto \left(\frac{d}{dx} f(x)\right) g(x) + f(x) \left(\frac{d}{dx} g(x)\right)\; . +\end{aligned} +\label{EquationMultiplicationRule} +\end{equation} + +When formulae are represented as data structures, symbolically differentiating an expression tree is a perfectly mechanistic process, considered subject to mechanical automation even at the very inception of calculus \citep{Leibniz1685}. This is realized in modern computer algebra systems such as Mathematica, Maxima, and Maple and machine learning frameworks such as Theano. + +In optimization, symbolic derivatives can give valuable insight into the structure of the problem domain and, in some cases, produce analytical solutions of extrema (e.g., solving for $\frac{d}{dx}f(x)=0$) that can eliminate the need for derivative calculation altogether. On the other hand, symbolic derivatives do not lend themselves to efficient runtime calculation of derivative values, as they can get exponentially larger than the expression whose derivative they represent. + +Consider a function $h(x)=f(x)g(x)$ and the multiplication rule in Eq.~\ref{EquationMultiplicationRule}. Since $h$ is a product, $h(x)$ and $\frac{d}{dx}h(x)$ have some common components, namely $f(x)$ and $g(x)$. Note also that on the right hand side, $f(x)$ and $\frac{d}{dx}f(x)$ appear separately. If we just proceeded to symbolically differentiate $f(x)$ and plugged its derivative into the appropriate place, we would have nested duplications of any computation that appears in common between $f(x)$ and $\frac{d}{dx}f(x)$. Hence, careless symbolic differentiation can easily produce exponentially large symbolic expressions which take correspondingly long to evaluate. This problem is known as \emph{expression swell} (Table~\ref{TableExpressionSwell}). + +\begin{table} + \centering + \renewcommand{\arraystretch}{1.2} + \caption{Iterations of the logistic map $l_{n+1}=4l_n (1-l_n)$, $l_1=x$ and the corresponding derivatives of $l_n$ with respect to $x$, illustrating expression swell.} + \label{TableExpressionSwell} + {\small + \begin{tabularx}{\columnwidth}{@{}lp{2.8cm}XX@{}} + \toprule + $n$ & $l_n$ & $\frac{d}{dx}l_n$ & $\frac{d}{dx}l_n$ (Simplified form)\\ + \addlinespace + \midrule + 1 & $x$ & $1$ & $1$\\ + \addlinespace + 2 & $4x(1 - x)$ & $4(1 - x) -4x$ & $4 - 8x$\\ + \addlinespace + 3 & $16x(1 - x)(1 - 2 x)^2$ & $16(1 - x)(1 - 2 x)^2 - 16x(1 - 2 x)^2 - 64x(1 - x)(1 - 2 x)$ & $16 (1 - 10 x + 24 x^2 - 16 x^3)$\\ + \addlinespace + 4 & $64x(1 - x)(1 - 2 x)^2$ $(1 - 8 x + 8 x^2)^2$ & $128x(1 - x)(-8 + 16 x)(1 - 2 x)^2 (1 - 8 x + 8 x^2) + 64 (1 - x)(1 - 2 x)^2 (1 - 8 x + 8 x^2)^2 - 64x(1 - 2 x)^2 (1 - 8 x + 8 x^2)^2 - 256x(1 - x)(1 - 2 x)(1 - 8 x + 8 x^2)^2$ & $64 (1 - 42 x + 504 x^2 - 2640 x^3 + 7040 x^4 - 9984 x^5 + 7168 x^6 - 2048 x^7)$\\ + \bottomrule + \end{tabularx}} +\end{table} + +When we are concerned with the accurate numerical evaluation of derivatives and not so much with their actual symbolic form, it is in principle possible to significantly simplify computations by storing only the values of intermediate sub-expressions in memory. Moreover, for further efficiency, we can interleave as much as possible the differentiation and simplification steps. This interleaving idea forms the basis of AD and provides an account of its simplest form: \emph{apply symbolic differentiation at the elementary operation level and keep intermediate numerical results, in lockstep with the evaluation of the main function.} This is AD in the forward accumulation mode, which we shall introduce in the following section. + + +\section{AD and Its Main Modes} +\label{SectionPreliminaries} + +AD can be thought of as performing a non-standard interpretation of a computer program where this interpretation involves augmenting the standard computation with the calculation of various derivatives. All numerical computations are ultimately compositions of a finite set of elementary operations for which derivatives are known \citep{Verma2000,Griewank2008}, and combining the derivatives of the constituent operations through the chain rule gives the derivative of the overall composition. Usually these elementary operations include the binary arithmetic operations, the unary sign switch, and transcendental functions such as the exponential, the logarithm, and the trigonometric functions. + +On the left hand side of Table~\ref{TableForwardADExample} we see the representation of the computation $y = f(x_1, x_2) = \ln(x_1) + x_1 x_2 - \sin(x_2)$ as an \emph{evaluation trace} of elementary operations---also called a Wengert list \citep{Wengert1964}. We adopt the three-part notation used by \citet{Griewank2008}, where a function $f: \mathbb{R}^n \to \mathbb{R}^m$ is constructed using intermediate variables $v_i$ such that +\begin{compactitem} + \item variables $v_{i-n} = x_i,\;i = 1, \dotsc, n$ are the input variables, + \item variables $v_i\;i = 1, \dotsc, l$ are the working (intermediate) variables, and + \item variables $y_{m-i} = v_{l-i},\;i = m - 1, \dotsc, 0$ are the output variables. +\end{compactitem} +Figure~\ref{FigureComputationalGraph} shows the given trace of elementary operations represented as a computational graph \citep{Bauer1974}, useful in visualizing dependency relations between intermediate variables. + +\begin{figure} + \centering + \trimbox{0cm -0.5cm}{\resizebox{0.8\textwidth}{!}{\normalsize\input{figures/comp-graph/comp-graph.tex}}} + \caption{Computational graph of the example $f(x_1, x_2) = \ln(x_1) + x_1 x_2 - \sin(x_2)$. See the primal trace in Tables \ref{TableForwardADExample} or \ref{TableReverseADExample} for the definitions of the intermediate variables $v_{-1} \dots v_5$\;.} + \label{FigureComputationalGraph} +\end{figure} + +Evaluation traces form the basis of the AD techniques. An important point to note here is that AD can differentiate not only closed-form expressions in the classical sense, but also algorithms making use of control flow such as branching, loops, recursion, and procedure calls, giving it an important advantage over symbolic differentiation which severely limits such expressivity. This is thanks to the fact that any numeric code will eventually result in a numeric evaluation trace with particular values of the input, intermediate, and output variables, which are the only things one needs to know for computing derivatives using chain rule composition, regardless of the specific control flow path that was taken during execution. Another way of expressing this is that AD is blind with respect to any operation, including control flow statements, which do not directly alter numeric values. + +\subsection{Forward Mode} + +AD in forward accumulation mode\footnote{Also called \emph{tangent linear} mode.} is the conceptually most simple type. Consider the evaluation trace of the function $f(x_1, x_2) = \ln(x_1) + x_1 x_2 - \sin(x_2)$ given on the left-hand side in Table~\ref{TableForwardADExample} and in graph form in Figure~\ref{FigureComputationalGraph}. For computing the derivative of $f$ with respect to $x_1$, we start by associating with each intermediate variable $v_i$ a derivative +\begin{equation*} + \dot{v}_i = \frac{\partial v_i}{\partial x_1}\; . +\end{equation*} + +Applying the chain rule to each elementary operation in the forward primal trace, we generate the corresponding tangent (derivative) trace, given on the right-hand side in Table~\ref{TableForwardADExample}. Evaluating the primals $v_i$ in lockstep with their corresponding tangents $\dot{v}_i$ gives us the required derivative in the final variable $\dot{v}_5=\frac{\partial y}{\partial x_1}$\;. + +%\footnote{In an implementation, the order in which $v_i$ and its corresponding $\dot{v}_i$ are computed can make a difference.} + +\begin{table} + \centering + \renewcommand{\arraystretch}{1.2} + \caption{Forward mode AD example, with $y = f(x_1, x_2) = \ln(x_1) + x_1 x_2 - \sin(x_2)$ evaluated at $(x_1, x_2) = (2, 5)$ and setting $\dot{x}_1 = 1$ to compute $\frac{\partial y}{\partial x_1}$. The original forward evaluation of the primals on the left is augmented by the tangent operations on the right, where each line complements the original directly to its left.} + \label{TableForwardADExample} + \begin{minipage}[c]{0.47\textwidth} + {\footnotesize + \begin{tabularx}{\textwidth}{p{0.2mm}p{2mm}p{16mm}X} + \toprule + \multicolumn{4}{l}{Forward Primal Trace}\\ + \multirow{9}{2mm}{\begin{tikzpicture}\draw[->,>=triangle 60,thick](0,0)--(0,-3.8);\end{tikzpicture}} & $v_{-1}$ & $=x_1$ & $=2$\\ + & $v_0$ & $=x_2$ & $=5$\\ + \cmidrule{2-4} + & $v_1$ & $=\ln{v_{-1}}$ & $=\ln{2}$\\ + & $v_2$ & $=v_{-1} \times v_0$ & $=2 \times 5$\\ + & $v_3$ & $=\sin{v_0}$ & $=\sin{5}$\\ + & $v_4$ & $=v_1+v_2$ & $=0.693+10$\\ + & $v_5$ & $=v_4-v_3$ & $=10.693+0.959$\\ + \cmidrule{2-4} + & $y$ & $=v_5$ & $=11.652$\\ + \bottomrule + \end{tabularx}} + \end{minipage} + \begin{minipage}[c]{0.52\textwidth} + \setlength{\fboxsep}{0pt}\colorbox{gray!20} + {\footnotesize + \begin{tabularx}{\textwidth}{p{0.2mm}p{2mm}p{28mm}X} + \toprule + \multicolumn{4}{l}{Forward Tangent (Derivative) Trace}\\ + \multirow{9}{2mm}{\begin{tikzpicture}\draw[->,>=triangle 60,thick](0,0)--(0,-3.8);\end{tikzpicture}} & $\dot{v}_{-1}$ & $=\dot{x}_1$ & $=1$\\ + & $\dot{v}_0$ & $=\dot{x}_2$ & $=0$\\ + \cmidrule{2-4} + & $\dot{v}_1$ & $=\dot{v}_{-1}/v_{-1}$ & $=1/2$\\ + & $\dot{v}_2$ & $=\dot{v}_{-1} \times v_0 + \dot{v}_0 \times v_{-1}$ & $=1 \times 5 + 0 \times 2$\\ + & $\dot{v}_3$ & $=\dot{v}_0 \times \cos{v_0}$ & $=0 \times \cos{5}$\\ + & $\dot{v}_4$ & $=\dot{v}_1+\dot{v}_2$ & $=0.5+5$\\ + & $\dot{v}_5$ & $=\dot{v}_4-\dot{v}_3$ & $=5.5-0$\\ + \cmidrule{2-4} + & \boldmath$\dot{y}$ & \boldmath$=\dot{v}_5$ & \boldmath$=5.5$\\ + \bottomrule + \end{tabularx}} + \end{minipage} +\end{table} + +This generalizes naturally to computing the Jacobian of a function $f : \mathbb{R}^n \to \mathbb{R}^m$ with $n$ independent (input) variables $x_i$ and $m$ dependent (output) variables $y_j$. In this case, each forward pass of AD is initialized by setting only one of the variables $\dot{x}_i=1$ and setting the rest to zero (in other words, setting $\dot{\mathbf{x}} = \mathbf{e}_i$, where $\mathbf{e}_i$ is the $i$-th unit vector). A run of the code with specific input values $\mathbf{x}=\mathbf{a}$ then computes +\begin{equation*} +\dot{y}_j = \left.\frac{\partial y_j}{\partial x_i}\right|_{\mathbf{x}=\mathbf{a}},\;j = 1, \dotsc, m\;, +\end{equation*} +giving us one column of the Jacobian matrix +\begin{equation*} +\mathbf{J}_f = \left. \begin{bmatrix} + \frac{\partial y_1}{\partial x_1} & \cdots & \frac{\partial y_1}{\partial x_n} \\ + \vdots & \ddots & \vdots \\ + \frac{\partial y_m}{\partial x_1} & \cdots & \frac{\partial y_m}{\partial x_n} + \end{bmatrix} \right|_{\mathbf{x}\; = \; \mathbf{a}} +\end{equation*} +evaluated at point $\mathbf{a}$. Thus, the full Jacobian can be computed in $n$ evaluations. + +Furthermore, forward mode AD provides a very efficient and matrix-free way of computing Jacobian--vector products +\begin{equation} + \mathbf{J}_f\,\mathbf{r} = \begin{bmatrix} + \frac{\partial y_1}{\partial x_1} & \cdots & \frac{\partial y_1}{\partial x_n} \\ + \vdots & \ddots & \vdots \\ + \frac{\partial y_m}{\partial x_1} & \cdots & \frac{\partial y_m}{\partial x_n} + \end{bmatrix} + \begin{bmatrix} + r_1 \\ + \vdots \\ + r_n + \end{bmatrix}\; , +\label{EquationJacobianVectorProduct} +\end{equation} +simply by initializing with $\dot{\mathbf{x}}=\mathbf{r}$. Thus, we can compute the Jacobian--vector product in just one forward pass. As a special case, when $f: \mathbb{R}^n \to \mathbb{R}$, we can obtain the directional derivative along a given vector $\mathbf{r}$ as a linear combination of the partial derivatives +\begin{equation*} + \nabla f \cdot \mathbf{r} +\end{equation*} +by starting the AD computation with the values $\dot{\mathbf{x}}=\mathbf{r}$. + +Forward mode AD is efficient and straightforward for functions $f: \mathbb{R} \to \mathbb{R}^m$, as all the derivatives $\frac{d y_i}{d x}$ can be computed with just one forward pass. Conversely, in the other extreme of $f: \mathbb{R}^n \to \mathbb{R}$, forward mode AD requires $n$ evaluations to compute the gradient +\begin{equation*} + \nabla f = \left( \frac{\partial y}{\partial x_1}, \dots, \frac{\partial y}{\partial x_n}\right)\; , +\end{equation*} +which also corresponds to a $1 \times n$ Jacobian matrix that is built one column at a time with the forward mode in $n$ evaluations. + +In general, for cases $f: \mathbb{R}^n \to \mathbb{R}^m$ where $n \gg m$, a different technique is often preferred. +We will describe AD in \emph{reverse accumulation mode} in Section~\ref{sec:reverse-mode}. + +\subsubsection{Dual Numbers} +\label{SectionDualNumbers} +Mathematically, forward mode AD (represented by the left- and right-hand sides in Table~\ref{TableForwardADExample}) can be viewed as evaluating a function using dual numbers,\footnote{First introduced by \citet{Clifford1873}, with important uses in linear algebra and physics.} which can be defined as truncated Taylor series of the form +\begin{equation*} + v + \dot{v}\epsilon \;, +\end{equation*} +where $v, \dot{v} \in \mathbb{R}$ and $\epsilon$ is a nilpotent number such +that $\epsilon^2 = 0$ and $\epsilon \neq 0$. Observe, for example, that +\begin{align*} + (v + \dot{v}\epsilon) + (u + \dot{u}\epsilon) &= (v + u) + (\dot{v} + \dot{u})\epsilon\\ + (v + \dot{v}\epsilon)(u + \dot{u}\epsilon) &= (vu) + (v\dot{u} + \dot{v}u)\epsilon\;, +\end{align*} +in which the coefficients of $\epsilon$ conveniently mirror symbolic differentiation rules (e.g., Eq.~\ref{EquationMultiplicationRule}). We can utilize this by setting up a regime where +\begin{equation}\label{EquationDualRule} + f(v + \dot{v}\epsilon) = f(v) + f'(v)\dot{v}\epsilon +\end{equation} +and using dual numbers as data structures for carrying the tangent value together with the primal.\footnote{Just as the complex number written $x + y i$ is represented in the computer as a pair in memory $(x, y)$ whose two slots are reals, the dual number written $x + \dot{x}\epsilon$ is represented as the pair $(x, \dot{x})$. Such pairs are sometimes called Argand pairs \citep[][p107 Eqs.~(157) and (158)]{Hamilton1837}.} The chain rule works as expected on this representation: two applications of Eq.~\ref{EquationDualRule} give +\begin{align*} + f(g(v + \dot{v}\epsilon)) &= f(g(v) + g'(v)\dot{v}\epsilon)\\ + &= f(g(v)) + f'(g(v))g'(v)\dot{v}\epsilon\;. +\end{align*} +The coefficient of $\epsilon$ on the right-hand side is exactly the derivative of the composition of $f$ and $g$. This means that since we implement elementary operations to respect the invariant Eq.~\ref{EquationDualRule}, all compositions of them will also do so. This, in turn, means that we can extract the derivative of a function by interpreting any non-dual number $v$ as $v + 0 \epsilon$ and evaluating the function in this non-standard way on an initial input with a coefficient $1$ for $\epsilon$: +\begin{align*} +\left.\frac{df(x)}{dx}\right|_{x=v} = \textrm{epsilon-coefficient}(\textrm{dual-version}(f)(v + 1\epsilon))\;. +\end{align*} + +This also extends to arbitrary program constructs, since dual numbers, as data types, can be contained in any data structure. As long as a dual number remains in a data structure with no arithmetic operations being performed on it, it will just remain a dual number; and if it is taken out of the data structure and operated on again, then the differentiation will continue. + +In practice, a function $f$ coded in a programming language of choice would be fed into an AD tool, which would then augment it with corresponding extra code to handle the dual operations so that the function and its derivative are simultaneously computed. This can be implemented through calls to a specific library, in the form of source code transformation where a given source code will be automatically modified, or through operator overloading, making the process transparent to the user. We discuss these implementation techniques in Section~\ref{SectionImplementations}. + +\subsection{Reverse Mode} +\label{sec:reverse-mode} + +AD in the reverse accumulation mode\footnote{Also called \emph{adjoint} or \emph{cotangent linear} mode.} corresponds to a generalized backpropagation algorithm, in that it propagates derivatives backward from a given output. This is done by complementing each intermediate variable $v_i$ with an adjoint +\begin{equation*} + \bar{v}_i = \frac{\partial y_j}{\partial v_i}\; , +\end{equation*} +which represents the sensitivity of a considered output $y_j$ with respect to changes in $v_i$. In the case of backpropagation, $y$ would be a scalar corresponding to the error $E$ (Figure~\ref{FigureBackpropagation}). + +In reverse mode AD, derivatives are computed in the second phase of a two-phase process. In the first phase, the original function code is run \emph{forward}, populating intermediate variables $v_i$ and recording the dependencies in the computational graph through a bookkeeping procedure. In the second phase, derivatives are calculated by propagating adjoints $\bar{v}_i$ in \emph{reverse}, from the outputs to the inputs. + +Returning to the example $y = f(x_1, x_2) = \ln(x_1) + x_1 x_2 - \sin(x_2)$, in Table~\ref{TableReverseADExample} we see the adjoint statements on the right-hand side, corresponding to each original elementary operation on the left-hand side. In simple terms, we are interested in computing the contribution $\bar{v}_i = \frac{\partial y}{\partial v_i}$ of the change in each variable $v_i$ to the change in the output $y$. Taking the variable $v_0$ as an example, we see in Figure~\ref{FigureComputationalGraph} that the only way it can affect $y$ is through affecting $v_2$ and $v_3$, so its contribution to the change in $y$ is given by +\begin{align*} + \frac{\partial y}{\partial v_0} &= \frac{\partial y}{\partial v_2}\frac{\partial v_2}{\partial v_0} + \frac{\partial y}{\partial v_3}\frac{\partial v_3}{\partial v_0}& + \text{or}&& + \bar{v}_0 &= \bar{v}_2\frac{\partial v_2}{\partial v_0} + \bar{v}_3\frac{\partial v_3}{\partial v_0}\;. +\end{align*} + +In Table~\ref{TableReverseADExample}, this contribution is computed in two incremental steps +\begin{align*} + \bar{v}_0 &= \bar{v}_3\frac{\partial v_3}{\partial v_0} & + \text{and} && + \bar{v}_0 &= \bar{v}_0 + \bar{v}_2\frac{\partial v_2}{\partial v_0}\;, +\end{align*} +lined up with the lines in the forward trace from which these expressions originate. + +After the forward pass on the left-hand side, we run the reverse pass of the adjoints on the right-hand side, starting with $\bar{v}_5 = \bar{y} = \frac{\partial y}{\partial y} = 1$. In the end we get the derivatives $\frac{\partial y}{\partial x_1} = \bar{x}_1$ and $\frac{\partial y}{\partial x_2} = \bar{x}_2$ in just one reverse pass. + +\begin{table} + \centering + \renewcommand{\arraystretch}{1.2} + \caption{Reverse mode AD example, with $y = f(x_1, x_2) = \ln(x_1) + x_1 x_2 - \sin(x_2)$ evaluated at $(x_1, x_2) = (2, 5)$. After the forward evaluation of the primals on the left, the adjoint operations on the right are evaluated in reverse (cf.\ Figure~\ref{FigureBackpropagation}). Note that both $\frac{\partial y}{\partial x_1}$ and $\frac{\partial y}{\partial x_2}$ are computed in the same reverse pass, starting from the adjoint $\bar{v}_5 = \bar{y} = \frac{\partial y}{\partial y} = 1$.} + \label{TableReverseADExample} + \begin{minipage}[t]{0.41\textwidth} + {\footnotesize + \begin{tabularx}{\textwidth}[t]{p{0.5mm}p{0.8mm}p{18mm}@{}X} + \toprule + \multicolumn{4}{l}{Forward Primal Trace}\\ + \multirow{9}{1mm}{\begin{tikzpicture}\draw[->,>=triangle 60,thick](0,0)--(0,-5.4);\end{tikzpicture}} & $v_{-1}$ & $=x_1$ & $=2$\\ + & $v_0$ & $=x_2$ & $=5$\\ + \cmidrule{2-4} + & $v_1$ & $=\ln{v_{-1}}$ & $=\ln{2}$\vspace{0.25mm}\\ + & $v_2$ & $=v_{-1} \times v_0$ & $=2 \times 5$\vspace{0.25mm}\\ + &\vspace{0.25mm}\\ + & $v_3$ & $=\sin{v_0}$ & $=\sin{5}$\vspace{0.25mm}\\ + & $v_4$ & $=v_1+v_2$ & $=0.693+10$\vspace{0.25mm}\\ + &\vspace{0.25mm}\\ + & $v_5$ & $=v_4-v_3$ & $=10.693+0.959$\vspace{0.25mm}\\ + &\vspace{0.25mm}\\ + \cmidrule{2-4} + & $y$ & $=v_5$ & $=11.652$\\ + \bottomrule + \end{tabularx}}\vspace{1mm} + \end{minipage} + \begin{minipage}[t]{0.58\textwidth} + \setlength{\fboxsep}{0pt}\colorbox{gray!20} + {\footnotesize + \begin{tabularx}{\textwidth}[t]{p{0.5mm}p{1mm}p{23mm}@{\hspace{1mm}}p{24mm}@{}X} + \toprule + \multicolumn{5}{l}{Reverse Adjoint (Derivative) Trace}\\ + \multirow{9}{1mm}{\begin{tikzpicture}\draw[<-,>=triangle 60,thick](0,0)--(0,-5.4);\end{tikzpicture}} & \boldmath$\bar{x}_1$ & \boldmath$=\bar{v}_{-1}$ & & \boldmath$=5.5$\\ + & \boldmath$\bar{x}_2$ & \boldmath$=\bar{v}_0$ & & \boldmath$=1.716$\\ + \cmidrule{2-5} + & $\bar{v}_{-1}$ & $=\bar{v}_{-1} + \bar{v}_1 \frac{\partial v_1}{\partial v_{-1}}$ & $=\bar{v}_{-1} + \bar{v}_1 / v_{-1}$ & $=5.5$\\ + & $\bar{v}_0$ & $=\bar{v}_0 + \bar{v}_2 \frac{\partial v_2}{\partial v_0}$ & $=\bar{v}_0 + \bar{v}_2 \times v_{-1}$ & $=1.716$\\ + & $\bar{v}_{-1}$ & $=\bar{v}_2 \frac{\partial v_2}{\partial v_{-1}}$ & $=\bar{v}_2 \times v_0$ & $=5$\\ + & $\bar{v}_0$ & $=\bar{v}_3 \frac{\partial v_3}{\partial v_0}$ & $=\bar{v}_3 \times \cos{v_0}$ & $=-0.284$\\ + & $\bar{v}_2$ & $=\bar{v}_4 \frac{\partial v_4}{\partial v_2}$ & $=\bar{v}_4 \times 1$ & $=1$\\ + & $\bar{v}_1$ & $=\bar{v}_4 \frac{\partial v_4}{\partial v_1}$ & $=\bar{v}_4 \times 1$ & $=1$\\ + & $\bar{v}_3$ & $=\bar{v}_5 \frac{\partial v_5}{\partial v_3}$ & $=\bar{v}_5 \times (-1)$ & $=-1$\\ + & $\bar{v}_4$ & $=\bar{v}_5 \frac{\partial v_5}{\partial v_4}$ & $=\bar{v}_5 \times 1$ & $=1$\\ + \cmidrule{2-5} + & $\bar{v}_5$ & $=\bar{y}$ & $=1$\\ + \bottomrule + \end{tabularx}} + \end{minipage} +\end{table} + +Compared with the straightforwardness of forward accumulation mode, reverse mode AD can, at first, appear somewhat ``mysterious'' \citep{Dennis1996}. \citet{Griewank2008} argue that this is in part because of the common acquaintance with the chain rule as a mechanistic procedure propagating derivatives forward. + +An important advantage of the reverse mode is that it is significantly less costly to evaluate (in terms of operation count) than the forward mode for functions with a large number of inputs. In the extreme case of $f: \mathbb{R}^n \to \mathbb{R}$, only one application of the reverse mode is sufficient to compute the full gradient $\nabla f = \left(\frac{\partial y}{\partial x_1},\dots,\frac{\partial y}{\partial x_n}\right)$, compared with the $n$ passes of the forward mode needed for populating the same. Because machine learning practice principally involves the gradient of a scalar-valued objective with respect to a large number of parameters, this establishes the reverse mode, as opposed to the forward mode, as the mainstay technique in the form of the backpropagation algorithm. + +In general, for a function $f: \mathbb{R}^n \to \mathbb{R}^m$, if we denote the operation count to evaluate the original function by $\textrm{ops}(f)$, the time it takes to calculate the $m \times n$ Jacobian by the forward mode is $n\;c\;\textrm{ops}(f)$, whereas the same computation can be done via reverse mode in $m\;c\;\textrm{ops}(f)$, where $c$ is a constant guaranteed to be $c<6$ and typically $c \sim [2,3]$ \citep{Griewank2008}. That is to say, reverse mode AD performs better when $m \ll n$. + +Similar to the matrix-free computation of Jacobian--vector products with forward mode (Eq.~\ref{EquationJacobianVectorProduct}), reverse mode can be used for computing the transposed Jacobian--vector product +\begin{equation*} + \mathbf{J}^{\intercal}_f\,\mathbf{r} = \begin{bmatrix} + \frac{\partial y_1}{\partial x_1} & \cdots & \frac{\partial y_m}{\partial x_1} \\ + \vdots & \ddots & \vdots \\ + \frac{\partial y_1}{\partial x_n} & \cdots & \frac{\partial y_m}{\partial x_n} + \end{bmatrix} + \begin{bmatrix} + r_1 \\ + \vdots \\ + r_m + \end{bmatrix}\;, +\end{equation*} +by initializing the reverse phase with $\bar{\mathbf{y}}=\mathbf{r}$. + +The advantages of reverse mode AD, however, come with the cost of increased storage requirements growing (in the worst case) in proportion to the number of operations in the evaluated function. It is an active area of research to improve storage requirements in implementations by using advanced methods such as checkpointing strategies and data-flow analysis \citep{Dauvergne2006,siskind2017divide}. + +\subsection{Origins of AD and Backpropagation} + +Ideas underlying AD date back to the 1950s \citep{Nolan1953,Beda1959}. Forward mode AD as a general method for evaluating partial derivatives was essentially discovered by \citet{Wengert1964}. It was followed by a period of relatively low activity, until interest in the field was revived in the 1980s mostly through the work of \citet{Griewank1989}, also supported by improvements in modern programming languages and the feasibility of an efficient reverse mode AD. + +Reverse mode AD and backpropagation have an intertwined history. The essence of the reverse mode, cast in a continuous-time formalism, is the Pontryagin maximum principle \citep{Rozonoer-Pontryagin-1959a, Boltyanskii-Gamkrelidze-Pontryagin-1960a}. This method was understood in the control theory community \citep{Bryson-1962a, Bryson-Ho-1969a} and cast in more formal terms with discrete-time variables topologically sorted in terms of dependency by \citet{Werbos-1974a}. Prior to Werbos, the work by \citet{linnainmaa1970representation,linnainmaa1976taylor} is often cited as the first published description of the reverse mode. \citet{Speelpenning80} subsequently introduced reverse mode AD as we know it, in the sense that he gave the first implementation that was actually automatic, accepting a specification of a computational process written in a general-purpose programming language and automatically performing the reverse mode transformation. + +Incidentally, \citet{Hecht1989} cites the work of \citet{Bryson-Ho-1969a} and \citet{Werbos-1974a} as the two earliest known instances of backpropagation. Within the machine learning community, the method has been reinvented several times, such as by \citet{Parker1985}, until it was eventually brought to fame by \citet{rumelhart1986learning} and the Parallel Distributed Processing (PDP) group. The PDP group became aware of Parker's work only after their own discovery; similarly, Werbos' work was not appreciated until it was found by Parker \citep{Hecht1989}. This tells us an interesting story of two highly interconnected research communities that have somehow also managed to stay detached during this foundational period. + +For a thorough review of the development of AD, we advise readers to refer to \citet{Rall2006}. Interested readers are highly recommended to read \citet{Griewank2012} for an investigation of the origins of the reverse mode and \citet{schmidhuber2015deep} for the same for backpropagation. + +\section{AD and Machine Learning} +\label{SectionDerivativesAndMachineLearning} + +In the following, we examine the main uses of derivatives in machine learning and report on a selection of works where general-purpose AD, as opposed to just backpropagation, has been successfully applied in a machine learning context. Areas where AD has seen use include optimization, neural networks, computer vision, natural language processing, and probabilistic inference. + +\subsection{Gradient-Based Optimization} + +Gradient-based optimization is one of the pillars of machine learning \citep{bottou2016optimization}. Given an objective function $f: \mathbb{R}^n \to \mathbb{R}$, classical gradient descent has the goal of finding (local) minima $\mathbf{w}^* = \argmin_{\mathbf{w}} f(\mathbf{w})$ via updates of the form $\Del \mathbf{w} = -\eta \nabla f$, where $\eta>0$ is a step size. Gradient-based methods make use of the fact that $f$ decreases steepest if one goes in the direction of the negative gradient. The convergence rate of gradient-based methods is usually improved by adaptive step-size techniques that adjust the step size $\eta$ on every iteration \citep{duchi2011adaptive,schaul2013no,kingma2015adam}. + +As we have seen, for large $n$, reverse mode AD provides a highly efficient method for computing gradients.\footnote{See \url{http://DiffSharp.github.io/DiffSharp/examples-gradientdescent.html} for an example of a general-purpose AD-based gradient descent routine using DiffSharp.} Figure~\ref{FigureHelmholtz} and Table~\ref{TableHelmholtz} demonstrate how gradient computation scales differently for forward and reverse mode AD and numerical differentiation, looking at the Helmholtz free energy function that has been used in AD literature for benchmarking gradient calculations \citep{Griewank1989,Griewank2008,griewank2012numerical}. + +\begin{figure} + \centering + \trimbox{0cm -1cm}{\resizebox{0.65\textwidth}{!}{\normalsize\input{figures/helmholtz/helmholtz.tex}}} + \caption{Evaluation time of the Helmholtz free energy function of a mixed fluid, based on the Peng-Robinson equation of state \citep{Peng1976}, ${f(\mathbf{x}) = R \, T \sum_{i = 0}^{n} \log \frac{x_i}{1 - \mathbf{b^T} \mathbf{x}} - \frac{\mathbf{x^T} \mathbf{A} \mathbf{x}}{\sqrt{8} \mathbf{b^T} \mathbf{x}} \log \frac{1 + (1 + \sqrt{2}) \mathbf{b^T} \mathbf{x}}{1 + (1 - \sqrt{2}) \mathbf{b^T} \mathbf{x}}}$, where $R$ is the universal gas constant, $T$ is the absolute temperature, $\mathbf{b} \in \mathbb{R}^n$ is a vector of constants, $\mathbf{A} \in \mathbb{R}^{n \times n}$ is a symmetric matrix of constants, and $\mathbf{x} \in \mathbb{R}^n$ is the vector of independent variables describing the system. The plots show the evaluation time of $f$ and the gradient $\nabla f$ with numerical differentiation (central difference), forward mode AD, and reverse mode AD, as a function of the number of variables $n$. Reported times are relative to the evaluation time of $f$ with $n=1$. The lower plot uses logarithmic scale for illustrating the behavior for small $n$. Numerical results are given in Table~\ref{TableHelmholtz}. (Code: {\small\url{http://DiffSharp.github.io/DiffSharp/misc/Benchmarks-h-grad-v0.5.7.fsx}})} + \label{FigureHelmholtz} +\end{figure} + +\begin{table} + \centering + \renewcommand{\arraystretch}{1.2} + \setlength{\tabcolsep}{1.85mm} + \caption{Evaluation times of the Helmholtz free energy function and its gradient (Figure~\ref{FigureHelmholtz}). Times are given relative to that of the original function with both (1) $n=1$ and (2) $n$ corresponding to each column. (For instance, reverse mode AD with $n=43$ takes approximately twice the time to evaluate relative to the original function with $n=43$.) Times are measured by averaging a thousand runs on a machine with Intel Core i7-4785T 2.20 GHz CPU and 16 GB RAM, using DiffSharp 0.5.7. The evaluation time for the original function with $n=1$ is 0.0023 ms.} + \label{TableHelmholtz} + {\small + \begin{tabularx}{\columnwidth}{@{}p{42.8mm}rrrrrrrr@{}} + \toprule + & \multicolumn{4}{l}{$n$, number of variables}\\ + \cmidrule(l){2-9} + & 1 & 8 & 15 & 22 & 29 & 36 & 43 & 50 \\ + \midrule + $f$, original\\ + \hspace{2mm} Relative $n=1$ & 1 & 5.12 & 14.51 & 29.11 & 52.58 & 84.00 & 127.33 & 174.44 \\ + $\nabla f$, numerical diff.\\ + \hspace{2mm} Relative $n=1$ & 1.08 & 35.55 & 176.79 & 499.43 & 1045.29 & 1986.70 & 3269.36 & 4995.96 \\ + \hspace{2mm} Relative $n$ in column & 1.08 & 6.93 & 12.17 & 17.15 & 19.87 & 23.64 & 25.67 & 28.63 \\ + $\nabla f$, forward AD\\ + \hspace{2mm} Relative $n=1$ & 1.34 & 13.69 & 51.54 & 132.33 & 251.32 & 469.84 & 815.55 & 1342.07\\ + \hspace{2mm} Relative $n$ in column & 1.34 & 2.66 & 3.55 & 4.54 & 4.77 & 5.59 & 6.40 & 7.69 \\ + $\nabla f$, reverse AD\\ + \hspace{2mm} Relative $n=1$ & 1.52 & 11.12 & 31.37 & 67.27 & 113.99 & 174.62 & 254.15 & 342.33 \\ + \hspace{2mm} Relative $n$ in column & 1.52 & 2.16 & 2.16 & 2.31 & 2.16 & 2.07 & 1.99 & 1.96 \\ + \bottomrule + \end{tabularx}} +\end{table} + +%Second order descent methods, Newton, Gauss-Newton, Levenberg-Marquardt +Second-order methods based on Newton's method make use of both the gradient $\nabla f$ and the Hessian $\mathbf{H}_f$, working via updates of the form $\Del \mathbf{w} = -\eta\,\mathbf{H}^{-1}_f \nabla f$ and providing significantly faster convergence \citep{Press2007}. AD provides a way of automatically computing the exact Hessian, enabling succinct and convenient general-purpose implementations.\footnote{See \url{http://DiffSharp.github.io/DiffSharp/examples-newtonsmethod.html} for an implementation of Newton's method with the full Hessian.} Newton's method converges in fewer iterations, but this comes at the cost of having to compute $\mathbf{H}_f$ in each iteration. In large-scale problems, the Hessian is usually replaced by a numerical approximation using first-order updates from gradient evaluations, giving rise to quasi-Newton methods. A highly popular such method is the BFGS\footnote{After Broyden–Fletcher–Goldfarb–Shanno, who independently discovered the method in the 1970s.} algorithm, together with its limited-memory variant L-BFGS \citep{Dennis1996}. On the other hand, Hessians arising in large-scale applications are typically sparse. This sparsity along with symmetry can be readily exploited by AD techniques such as computational graph elimination \citep{Dixon1991}, partial separability \citep{Gay1996}, and matrix coloring and compression \citep{Gebremedhin2009}. + +In many cases one does not need the full Hessian but only a Hessian--vector product $\mathbf{H} \mathbf{v}$, which can be computed efficiently using a reverse-on-forward configuration of AD by applying the reverse mode to take the gradient of code produced by the forward mode.\footnote{\citet{Christianson2012ALN} demonstrates that the second derivative can be computed with the same arithmetic operation sequence using forward-on-reverse, reverse-on-forward, and reverse-on-reverse. The taping overheads of these methods may differ in implementation-dependent ways.} Given the function $f:\mathbb{R}^n\to\mathbb{R}$, the evaluation point $\mathbf{x}$, and the vector $\mathbf{v}$, one can accomplish this by first computing the directional derivative $\nabla f \cdot \mathbf{v}$ through the forward mode via setting $\dot{\mathbf{x}}=\mathbf{v}$ and then applying the reverse mode on this result to get $\nabla^2 f \cdot \mathbf{v}=\mathbf{H}_f \mathbf{v}$ \citep{Pearlmutter1994}. This computes $\mathbf{H} \mathbf{v}$ with $O(n)$ complexity, even though $\mathbf{H}$ is a $n \times n$ matrix. Availability of robust AD tools may make more sophisticated optimization methods applicable to large-scale machine-learning problems. For instance, when fast stochastic Hessian--vector products are available, these can be used as the basis of stochastic Newton's methods \citep{agarwal2016second}, which have the potential to endow stochastic optimization with quadratic convergence. + +Another approach for improving the rate of convergence of gradient-based methods is to use gain adaptation methods such as stochastic meta-descent (SMD) \citep{Schraudolph1999}, where stochastic sampling is introduced to avoid local minima and reduce the computational expense. An example using SMD with AD Hessian--vector products is given by \citet{Vishwanathan2006} on conditional random fields (CRF). Similarly, \citet{Schraudolph2003} use Hessian--vector products in their model combining conjugate gradient techniques with stochastic gradient descent. + +\subsection{Neural Networks, Deep Learning, Differentiable Programming} + +Training of a neural network is an optimization problem with respect to its set of weights, which can in principle be addressed by using any method ranging from evolutionary algorithms \citep{such2017deep} to gradient-based methods such as BFGS \citep{Apostolopoulou2009} or the mainstay stochastic gradient descent \citep{bottou2010large} and its many variants \citep{kingma2015adam,tieleman2012lecture,duchi2011adaptive}. As we have seen, the backpropagation algorithm is only a special case of AD: by applying reverse mode AD to an objective function evaluating a network's error as a function of its weights, we can readily compute the partial derivatives needed for performing weight updates.\footnote{See \url{http://DiffSharp.github.io/DiffSharp/examples-neuralnetworks.html} for an implementation of backpropagation with reverse mode AD.} + +The LUSH system \citep{LUSH2002}, and its predecessor SN \citep{bottou-lecun-88}, were the first production systems that targeted efficient neural network simulation while incorporating both a general-purpose programming language and AD. Modern deep learning frameworks provide differentiation capability in one way or another, but the underlying mechanism is not always made clear and confusion abounds regarding the use of the terms ``autodiff'', ``automatic differentiation'', and ``symbolic differentiation'', which are sometimes even used interchangeably. In mainstream frameworks including Theano\footnote{Theano is a computational graph optimizer and compiler with GPU support and it currently handles derivatives in a highly optimized form of symbolic differentiation. The result can be interpreted as a hybrid of symbolic differentiation and reverse mode AD, but Theano does not use the general-purpose reverse accumulation as we describe in this paper. (Personal communication with the authors.)} \citep{Bastien2012}, TensorFlow \citep{abadi2016tensorflow}, Caffe \citep{jia2014caffe}, and CNTK \citep{seide2016cntk} the user first constructs a model as a computational graph using a domain-specific mini language, which then gets interpreted by the framework during execution. This approach has the advantage of enabling optimizations of the computational graph structure (e.g., as in Theano), but the disadvantages of having limited and unintuitive control flow and being difficult to debug. In contrast, the lineage of recent frameworks led by autograd \citep{maclaurin2016modeling}, Chainer \citep{tokui2015chainer}, and PyTorch \citep{paszke2017automatic} provide truly general-purpose reverse mode AD of the type we outline in Section~\ref{SectionPreliminaries}, where the user directly uses the host programming language to define the model as a regular program of the forward computation. This eliminates the need for an interpreter, allows arbitrary control flow statements, and makes debugging simple and intuitive. + +Simultaneously with the ongoing adoption of general-purpose AD in machine learning, we are witnessing a modeling-centric terminology emerge within the deep learning community. The terms \emph{define-and-run} and \emph{static computational graph} refer to Theano-like systems where a model is constructed, before execution, as a computational graph structure, which later gets executed with different inputs while remaining fixed. In contrast, the terms \emph{define-by-run} and \emph{dynamic computational graph} refer to the general-purpose AD capability available in newer PyTorch-like systems where a model is a regular program in the host programming language, whose execution dynamically constructs a computational graph on-the-fly that can freely change in each iteration.\footnote{Note that the terms ``static'' and ``dynamic'' here are used in the sense of having a fixed versus non-fixed computational graph topology and not in the sense of data flow architectures.} + +\emph{Differentiable programming}\footnote{A term advocated by Christopher Olah (\url{http://colah.github.io/posts/2015-09-NN-Types-FP/}), David Dalrymple (\url{https://www.edge.org/response-detail/26794}), and Yann LeCun (\url{https://www.facebook.com/yann.lecun/posts/10155003011462143}) from a deep learning point of view. Note the difference from \emph{differential} dynamic programming \citep{jacobson1970differential} in optimal control.} is another emerging term referring to the realization that deep learning practice essentially amounts to writing program templates of potential solutions to a problem, which are constructed as differentiable directed graphs assembled from functional blocks whose parameters are learned from examples using gradient-based optimization. Expressed in this paradigm, neural networks are just a class of parameterized differentiable programs composed of building blocks such as feed-forward, convolutional, and recurrent elements. We are increasingly seeing these traditional building blocks freely composed in arbitrary algorithmic structures using control flow, as well as the introduction of novel differentiable architectures such as the neural Turing machine \citep{graves2014neural}, a range of controller--interface abstractions \citep{graves2016hybrid,zaremba2016learning,joulin2015inferring,sukhbaatar2015end}, and differentiable versions of data structures such as stacks, queues, deques \citep{grefenstette2015learning}. Availability of general-purpose AD greatly simplifies the implementation of such architectures by enabling their expression as regular programs that rely on the differentiation infrastructure. Although the differentiable programming perspective on deep learning is new, we note that programming with differentiable functions and having differentiation as a language infrastructure has been the main research subject of the AD community for many decades and realized in a wide range of systems and languages as we shall see in Section~\ref{SectionImplementations}. + +There are instances in neural network literature---albeit few---where explicit reference has been made to AD for computing error gradients, such as \citet{Eriksson1998} using AD for large-scale feed-forward networks, and the work by \citet{Yang2008}, where the authors use AD to train a neural-network-based proportional-integral-derivative (PID) controller. Similarly, \citet{Rollins2009} uses reverse mode AD in conjunction with neural networks for the problem of optimal feedback control. Another example is given for continuous time recurrent neural networks (CTRNN) by \citet{AlSeyab2008}, where the authors apply AD for the training of CTRNNs predicting dynamic behavior of nonlinear processes in real time and report significantly reduced training time compared with other methods. + + +\subsection{Computer Vision} + +Since the influential work by \citet{krizhevsky2012imagenet}, computer vision has been dominated by deep learning, specifically, variations of convolutional neural networks \citep{lecun1998gradient}. These models are trained end-to-end, meaning that a mapping from raw input data to corresponding outputs is learned, automatically discovering the representations needed for feature detection in a process called representation learning \citep{bengio2013representation}. + +Besides deep learning, an interesting area where AD can be applied to computer vision problems is inverse graphics \citep{horn1977understanding,hinton1997generative}---or analysis-by-synthesis \citep{yildirim2015efficient}---where vision is seen as the inference of parameters for a generative model of a scene. Using gradient-based optimization in inverse graphics requires propagating derivatives through whole image synthesis pipelines including the renderer. \citet{eslami2016attend} use numerical differentiation for this purpose. \citet{loper2014opendr} implement the Open Differentiable Renderer (OpenDR), which is a scene renderer that also supplies derivatives of the image pixels with respect to scene parameters, and demonstrate it in the task of fitting an articulated and deformable 3D model of the human body to image and range data from a Kinect device. Similarly, \citet{kulkarni2015picture} implement a differentiable approximate renderer for the task of inference in probabilistic programs describing scenes. + +\citet{srajer2016benchmark} investigate the use of AD for three tasks in computer vision and machine learning, namely bundle adjustment \citep{triggs1999bundle}, Gaussian mixture model fitting, and hand tracking \citep{taylor2014user}, and provide a comprehensive benchmark of various AD tools for the computation of derivatives in these tasks. + +\citet{Pock2007} make use of AD in addressing the problems of denoising, segmentation, and recovery of information from stereoscopic image pairs, and note the usefulness of AD in identifying sparsity patterns in large Jacobian and Hessian matrices. In another study, \citet{Grabner2008} use reverse mode AD for GPU-accelerated medical 2D/3D registration, a task involving the alignment of data from different sources such as X-ray images or computed tomography. The authors report a six-fold increase in speed compared with numerical differentiation using center difference (cf.\ our benchmark with the Helmholtz function, Figure~\ref{FigureHelmholtz} and Table~\ref{TableHelmholtz}). + +\citet{Barrett2013} present a use of general-purpose AD for the task of video event detection using hidden Markov models (HMMs) and \citet{Dalal2005} object detectors, performing training on a corpus of pre-tracked video using an adaptive step size gradient descent with reverse mode AD. Initially implemented with the R6RS-AD package\footnote{\url{https://github.com/qobi/R6RS-AD}} which provides forward and reverse mode AD in Scheme, the resulting gradient code was later ported to C and highly optimized.\footnote{Personal communication.} + + +\subsection{Natural Language Processing} + +Natural language processing (NLP) constitutes one of the areas where rapid progress is being made by applying deep learning techniques \citep{goldberg2016primer}, with applications in tasks including machine translation \citep{bahdanau2014neural}, language modeling \citep{mikolov2010recurrent}, dependency parsing \citep{chen2014fast}, and question answering \citep{pmlr-v48-kumar16}. Besides deep learning approaches, statistical models in NLP are commonly trained using general purpose or specialized gradient-based methods and mostly remain expensive to train. Improvements in training time can be realized by using online or distributed training algorithms \citep{Gimpel2010}. An example using stochastic gradient descent for NLP is given by \citet{Finkel2008} optimizing conditional random field parsers through an objective function. Related with the work on video event detection in the previous section, \citet{Yu2013} report their work on sentence tracking, representing an instance of grounded language learning paired with computer vision, where the system learns word meanings from short video clips paired with descriptive sentences. The method uses HMMs to represent changes in video frames and meanings of different parts of speech. This work is implemented in C and computes the required gradients using AD through the ADOL-C tool.\footnote{An implementation of the sentence tracker applied to video search using sentence-based queries can be accessed online: \url{http://upplysingaoflun.ecn.purdue.edu/~qobi/cccp/sentence-tracker-video-retrieval.html}} + +\subsection{Probabilistic Modeling and Inference} + +Inference in probabilistic models can be static, such as compiling a given model to Bayesian networks and using algorithms such as belief propagation for inference; or they can be dynamic, executing a model forward many times and computing statistics on observed values to infer posterior distributions. Markov chain Monte Carlo (MCMC) \citep{Neal1993} methods are often used for dynamic inference, such as the Metropolis--Hastings algorithm based on random sampling \citep{Chib1995}. \citet{Meyer2003} give an example of how AD can be used to speed up Bayesian posterior inference in MCMC, with an application in stochastic volatility. Amortized inference \citep{gershman2014amortized,stuhlmuller2013learning} techniques based on deep learning \citep{le2016inference,ritchie2016deep} work by training neural networks for performing approximate inference in generative models defined as probabilistic programs \citep{gordon2014probabilistic}. + +When model parameters are continuous, the Hamiltonian---or, hybrid---Monte Carlo (HMC) algorithm provides improved convergence characteristics avoiding the slow exploration of random sampling, by simulating Hamiltonian dynamics through auxiliary ``momentum variables'' \citep{Duane1987}. The advantages of HMC come at the cost of requiring gradient evaluations of complicated probability models. AD is highly suitable here for complementing probabilistic modeling, because it relieves the user from the manual derivation of gradients for each model.\footnote{See \url{http://diffsharp.github.io/DiffSharp/examples-hamiltonianmontecarlo.html} for an implementation of HMC with reverse mode AD.} For instance, the probabilistic programming language Stan \citep{carpenter2016stan} implements automatic Bayesian inference based on HMC and the No-U-Turn sampler (NUTS) \citep{Hoffman2014} and uses reverse mode AD for the calculation of gradients for both HMC and NUTS \citep{carpenter2015stan}. Similarly, \citet{Wingate2011} demonstrate the use of AD as a non-standard interpretation of probabilistic programs enabling efficient inference algorithms. \citet{kucukelbir2017automatic} present an AD-based method for deriving variational inference (VI) algorithms. + +PyMC3 \citep{salvatier2016probabilistic} allows fitting of Bayesian models using MCMC and VI, for which it uses gradients supplied by Theano. Edward \citep{tran2016edward} is a library for deep probabilistic modeling, inference, and criticism \citep{tran2017deep} that supports VI using TensorFlow. Availability of general-purpose AD in this area has enabled new libraries such as Pyro\footnote{\url{http://pyro.ai/}} and ProbTorch \citep{siddharth2017learning} for deep \emph{universal} probabilistic programming with support for recursion and control flow, relying, in both instances, on VI using gradients supplied by PyTorch's reverse mode AD infrastructure. + +When working with probabilistic models, one often needs to backpropagate derivatives through sampling operations of random variables in order to achieve stochastic optimization of model parameters. The score-function estimator, or REINFORCE \citep{williams1992simple}, method provides a generally applicable unbiased gradient estimate, albeit with high variance. When working with continuous random variables, one can substitute a random variable by a deterministic and differentiable transformation of a simpler random variable, a method known as the ``reparameterization trick'' \citep{williams1992simple,kingma2014auto,rezende2014stochastic}. For discrete variables, the REBAR \citep{tucker2017rebar} method provides a lower-variance unbiased gradient estimator by using continuous relaxation. A generalization of REBAR called RELAX \citep{grathwohl2017backpropagation} works by learning a free-form control variate parameterized by a neural network and is applicable in both discrete and continuous settings. + +\section{Implementations} +\label{SectionImplementations} + +It is useful to have an understanding of the different ways in which AD can be implemented. Here we cover major implementation strategies and provide a survey of existing tools. + +A principal consideration in any AD implementation is the performance overhead introduced by the AD arithmetic and bookkeeping. In terms of computational complexity, AD guarantees that the amount of arithmetic goes up by no more than a small constant factor \citep{Griewank2008}. On the other hand, managing this arithmetic can introduce a significant overhead if done carelessly. For instance, naïvely allocating data structures for holding dual numbers will involve memory access and allocation for every arithmetic operation, which are usually more expensive than arithmetic operations on modern computers.\footnote{The implementation of forward mode in Julia \citep{revels2016forward} attempts to avoid this, and some current compilers can avoid this expense by unboxing dual numbers \citep{leroy1997effectiveness, jones1993glasgow, jones1991unboxed, siskind2016efficient}. This method is also used to reduce the memory-access overhead in the implementations of forward mode in Stalingrad and the Haskell \emph{ad} library.} Likewise, using operator overloading may introduce method dispatches with attendant costs, which, compared to raw numerical computation of the original function, can easily amount to a slowdown of an order of magnitude.\footnote{Flow analysis \citep{shivers1991control} and/or partial evaluation \citep{jones1993partial}, together with tag stripping \citep{appel1989runtime, peterson1989untagged}, can remove this method dispatch. These, together with unboxing, can often make it possible to completely eliminate the memory access, memory allocation, memory reclamation, and method dispatch overhead of dual numbers \citep{siskind2016efficient}.} + +Another major issue is the risk of hitting a class of bugs called ``perturbation confusion'' \citep{SiskindPearlmutter2005a,manzyuk2012confusion}. This essentially means that if two ongoing differentiations affect the same piece of code, the two formal epsilons they introduce (Section~\ref{SectionDualNumbers}) need to be kept distinct. It is very easy to have bugs---particularly in performance-oriented AD implementations---that confuse these in various ways. Such situations can also arise when AD is nested, that is, derivatives are computed for functions that internally compute derivatives. + +Translation of mathematics into computer code often requires attention to numeric issues. For instance, the mathematical expressions $\log(1 + x)$ or $\sqrt{x^2+y^2+z^2}$ or $\tan^{-1}(y/x)$ should not be naïvely translated, but rather expressed as \texttt{log1p(x)}, \texttt{hypot(x,hypot(y,z))}, and \texttt{atan2(y,x)}. In machine learning, the most prominent example of this is probably the so-called log-sum-exp trick to improve the numerics of calculations of the form $\log\sum_i \exp x_i$. AD is not immune to such numeric considerations. For example, code calculating $E=\sum_i E_i$, processed by AD, will calculate $\nabla_w E=\sum_i\nabla_w E_i$. If the system is seeking a local minimum of $E$ then $\nabla_w E = \sum_i\nabla_w E_i \rightarrow_t 0$, and naïvely adding a set of large numbers whose sum is near zero is numerically fraught. This is to say that AD is not immune to the perils of floating point arithmetic, and can sometimes introduce numeric issues which were not present in the primal calculation. Issues of numeric analysis are outside our present scope, but there is a robust literature on the numerics of AD (e.g., \citet{griewank2012numerical}) involving using subgradients to allow optimization to proceed despite non-differentiability of the objective, appropriate subgradients and approximations for functions like $\lvert \cdot \rvert$ and $\lVert\cdot\rVert_2$ and $\sqrt{\cdot}$ near zero, and a spate of related issues. + +One should also be cautious about approximated functions and AD \citep{sirkes-tziperman-1997a}. In this case, if one has a procedure \emph{approximating} an ideal function, AD always gives the derivative of the procedure that was actually programmed, which may not be a good approximation of the derivative of the ideal function that the procedure was approximating. For instance, consider $e^x$ computed by a piecewise-rational approximation routine. Using AD on this routine would produce an approximated derivative in which each piece of the piecewise formula will get differentiated. Even if this would remain an approximation of the derivative of $e^x$, we know that $\frac{de^x}{dx} = e^x$ and the original approximation itself was already a better approximation for the derivative of $e^x$.\footnote{In modern systems this is not an issue, because $e^x$ is a primitive implemented in hardware.} Users of AD implementations must be therefore cautious to \emph{approximate the derivative, not differentiate the approximation}. This would require explicitly approximating a known derivative, in cases where a mathematical function can only be computed approximately but has a well-defined mathematical derivative. + +We note that there are similarities as well as differences between machine learning workloads and those studied in the traditional AD literature \citep{baydin2016tricks}. Deep learning systems are generally compute-bound and spend a considerable amount of computation time in highly-optimized numerical kernels for matrix operations \citep{hadjis2015caffe,chetlur2014cudnn}. This is a situation which is arguably amenable to operator-overloading-based AD implementations on high-level operations, as is commonly found in current machine learning frameworks. In contrast, numerical simulation workloads in traditional AD applications can be bandwidth-bound, making source code transformation and compiler optimization approaches more relevant. Another difference worth noting is that whereas high numerical precision is desirable in traditional application domains of AD such as computational fluid dynamics \citep{cohen2009fast}, in deep learning lower-precision is sufficient and even desirable in improving computational efficiency, thanks to the error resiliency of neural networks \citep{gupta2015deep,courbariaux2015binaryconnect}. + +There are instances in recent literature where implementation-related experience from the AD field has been put to use in machine learning settings. One particular area of recent interest is implicit and iterative AD techniques \citep{Griewank2008}, which has found use in work incorporating constrained optimization within deep learning \citep{amos2017optnet} and probabilistic graphical models and neural networks \citep{johnson2016composing}. Another example is checkpointing strategies \citep{Dauvergne2006,siskind2017divide}, which allow balancing of application-specific trade-offs between time and space complexities of reverse mode AD by not storing the full tape of intermediate variables in memory and reconstructing these as needed by re-running parts of the forward computation from intermediate checkpoints. This is highly relevant in deep learning workloads running on GPUs with limited memory budgets. A recent example in this area is the work by \citet{gruslys2016memory}, where the authors construct a checkpointing variety of the backpropagation through time (BPTT) algorithm for recurrent neural networks and demonstrate it saving up to 95\% memory usage at the cost of a 33\% increase in computation time in one instance. + +In Table~\ref{TableADImplementations} we present a review of notable general-purpose AD implementations.\footnote{Also see the website \url{http://www.autodiff.org/} for a list of tools maintained by the AD community.} A thorough taxonomy of implementation techniques was introduced by \citet{Juedes1991}, which was later revisited by \citet{Bischof2008} and simplified into \emph{elemental}, \emph{operator overloading}, \emph{compiler-based}, and \emph{hybrid} methods. We adopt a similar classification for the following part of this section. + +% Remarks concerning further implementation methods in \citet{Gay2006}. + +\addtolength{\tabcolsep}{-3pt} +\begin{sidewaystable} + \centering + \renewcommand{\arraystretch}{1.5} + \caption{Survey of AD implementations. Tools developed primarily for machine learning are highlighted in bold.} + \label{TableADImplementations} + {\tiny + \begin{tabularx}{\textwidth}{@{}p{12mm}p{20mm}p{5mm}p{6mm}p{62mm}p{38mm}p{65mm}@{}} + \toprule + Language & Tool & Type & Mode & Institution / Project & Reference & URL\\ + \midrule + AMPL & AMPL & INT & F, R & Bell Laboratories & \citet{Fourer2002} & \tiny\url{http://www.ampl.com/}\\ + C, C++ & ADIC & ST & F, R & Argonne National Laboratory & \citet{Bischof1997} & \tiny\url{http://www.mcs.anl.gov/research/projects/adic/}\\ + & ADOL-C & OO & F, R & Computational Infrastructure for Operations Research & \citet{Walther2012} & \tiny\url{https://projects.coin-or.org/ADOL-C}\\ + C++ & Ceres Solver & LIB & F & Google & & \tiny\url{http://ceres-solver.org/}\\ + & CppAD & OO & F, R & Computational Infrastructure for Operations Research & \citet{Bell2008} & \tiny\url{http://www.coin-or.org/CppAD/}\\ + & FADBAD++ & OO & F, R & Technical University of Denmark & \citet{Bendtsen1996} & \tiny\url{http://www.fadbad.com/fadbad.html}\\ + & Mxyzptlk & OO & F & Fermi National Accelerator Laboratory & \citet{Ostiguy2007} & \\ + C\# & AutoDiff & LIB & R & George Mason Univ., Dept. of Computer Science & \citet{Shtof2013} & \tiny\url{http://autodiff.codeplex.com/}\\ + F\#, C\# & \textbf{DiffSharp} & OO & F, R & Maynooth University, Microsoft Research Cambridge & \citet{baydin2016diffsharp} & \tiny\url{http://diffsharp.github.io}\\ + Fortran & ADIFOR & ST & F, R & Argonne National Laboratory & \citet{Bischof1996} & \tiny\url{http://www.mcs.anl.gov/research/projects/adifor/}\\ + & NAGWare & COM & F, R & Numerical Algorithms Group & \citet{Naumann2005} & \tiny\url{http://www.nag.co.uk/nagware/Research/ad_overview.asp}\\ + & TAMC & ST & R & Max Planck Institute for Meteorology & \citet{Giering1998} & \tiny\url{http://autodiff.com/tamc/}\\ + Fortran, C & COSY & INT & F & Michigan State Univ., Biomedical and Physical Sci. & \citet{Berz1996} & \tiny\url{http://www.bt.pa.msu.edu/index_cosy.htm}\\ + & Tapenade & ST & F, R & INRIA Sophia-Antipolis & \citet{Hascoet2013} & \tiny\url{http://www-sop.inria.fr/tropics/tapenade.html}\\ + Haskell & ad & OO & F, R & Haskell package & & \tiny\url{http://hackage.haskell.org/package/ad}\\ + Java & ADiJaC & ST & F, R & University Politehnica of Bucharest & \citet{slusanschi2016adijac} & \tiny\url{http://adijac.cs.pub.ro}\\ + & Deriva & LIB & R & Java \& Clojure library & & \tiny\url{https://github.com/lambder/Deriva}\\ + Julia & JuliaDiff & OO & F, R & Julia packages & \citet{RevelsLubinPapamarkou2016} & \tiny\url{http://www.juliadiff.org/}\\ + Lua & \textbf{torch-autograd} & OO & R & Twitter Cortex & & \tiny\url{https://github.com/twitter/torch-autograd}\\ + MATLAB & ADiMat & ST & F, R & Technical University of Darmstadt, Scientific Comp. & \citet{Willkomm2013} & \tiny\url{http://adimat.sc.informatik.tu-darmstadt.de/}\\ + & INTLab & OO & F & Hamburg Univ. of Technology, Inst. for Reliable Comp. & \citet{Rump1999} & \tiny\url{http://www.ti3.tu-harburg.de/rump/intlab/}\\ + & TOMLAB/MAD & OO & F & Cranfield University \& Tomlab Optimization Inc. & \citet{Forth2006} & \tiny\url{http://tomlab.biz/products/mad}\\ + Python & ad & OO & R & Python package & & \tiny\url{https://pypi.python.org/pypi/ad}\\ + & \textbf{autograd} & OO & F, R & Harvard Intelligent Probabilistic Systems Group & \citet{maclaurin2016modeling} & \tiny\url{https://github.com/HIPS/autograd}\\ + & \textbf{Chainer} & OO & R & Preferred Networks & \citet{tokui2015chainer} & \tiny\url{https://chainer.org/}\\ + & \textbf{PyTorch} & OO & R & PyTorch core team & \citet{paszke2017automatic} & \tiny\url{http://pytorch.org/}\\ + & \textbf{Tangent} & ST & F, R & Google Brain & \citet{van2017tangent} & \tiny\url{https://github.com/google/tangent}\\ + Scheme & R6RS-AD & OO & F, R & Purdue Univ., School of Electrical and Computer Eng. & & \tiny\url{https://github.com/qobi/R6RS-AD}\\ + & Scmutils & OO & F & MIT Computer Science and Artificial Intelligence Lab. & \citet{Sussman2001} & \tiny\url{http://groups.csail.mit.edu/mac/users/gjs/6946/refman.txt}\\ + & Stalingrad & COM & F, R & Purdue Univ., School of Electrical and Computer Eng. & \citet{pearlmutter2008reverse} & \tiny\url{http://www.bcl.hamilton.ie/~qobi/stalingrad/}\\ + % Odyssee reverse mode source transformation (predecessor of Tapenade) + % TAMC reverse mode source transformation + \bottomrule + \addlinespace + \multicolumn{7}{l}{F: Forward, R: Reverse; COM: Compiler, INT: Interpreter, LIB: Library, OO: Operator overloading, ST: Source transformation} + \end{tabularx} + } +\end{sidewaystable} +\addtolength{\tabcolsep}{3pt} + +\subsection{Elemental Libraries} + +These implementations form the most basic category and work by replacing mathematical operations with calls to an AD-enabled library. Methods exposed by the library are then used in function definitions, meaning that the decomposition of any function into elementary operations is done manually when writing the code. + +The approach has been utilized since the early days of AD, with prototypical examples being the WCOMP and UCOMP packages of \citet{Lawson1971}, the APL package of \citet{Neidinger1989}, and the work by \citet{Hinkins1994}. Likewise, \citet{Hill1992} formulate their implementation of AD in MATLAB using elemental methods. + +Elemental libraries still constitute the simplest strategy to implement AD for languages without operator overloading. + +\subsection{Compilers and Source Code Transformation} + +These implementations provide extensions to programming languages that automate the decomposition of algorithms into AD-enabled elementary operations. They are typically executed as preprocessors\footnote{Preprocessors transform program source code before it is given as an input to a compiler.} to transform the input in the extended language into the original language. + +Classical instances of source code transformation include the Fortran preprocessors GRESS \citep{Horwedel1988} and PADRE2 \citep{Kubo1990}, which transform AD-enabled variants of Fortran into standard Fortran 77 before compiling. Similarly, the ADIFOR tool \citep{Bischof1996}, given a Fortran source code, generates an augmented code in which all specified partial derivatives are computed in addition to the original result. For procedures coded in ANSI C, the ADIC tool \citep{Bischof1997} implements AD as a source code transformation after the specification of dependent and independent variables. A recent and popular tool also utilizing this approach is Tapenade \citep{Pascual2008,Hascoet2013}, implementing forward and reverse mode AD for Fortran and C programs. Tapenade itself is implemented in Java and can be run locally or as an online service.\footnote{\url{http://www-tapenade.inria.fr:8080/tapenade/index.jsp}} + +In addition to language extensions through source code transformation, there are implementations introducing new languages with tightly integrated AD capabilities through special-purpose compilers or interpreters. Some of the earliest AD tools such as SLANG \citep{Adamson1969} and PROSE \citep{Pfeiffer1987} belong to this category. The NAGWare Fortran 95 compiler \citep{Naumann2005} is a more recent example, where the use of AD-related extensions triggers automatic generation of derivative code at compile time. + +As an example of interpreter-based implementation, the algebraic modeling language AMPL \citep{Fourer2002} enables objectives and constraints to be expressed in mathematical notation, from which the system deduces active variables and arranges the necessary AD computations. Other examples in this category include the FM/FAD package \citep{Mazourik1991}, based on the Algol-like DIFALG language, and the object-oriented COSY language \citep{Berz1996} similar to Pascal. + +The Stalingrad compiler \citep{pearlmutter2008reverse,Siskind2008}, working on the Scheme-based AD-aware VLAD language, also falls under this category. The newer DVL compiler\footnote{\url{https://github.com/axch/dysvunctional-language}} is based on Stalingrad and uses a reimplementation of portions of the VLAD language. + +Motivated by machine learning applications, the Tangent library \citep{van2017tangent} implements AD using source code transformation, and accepts numeric functions written in a syntactic subset of Python and Numpy. + +\subsection{Operator Overloading} + +In modern programming languages with polymorphic features, operator overloading provides the most straightforward way of implementing AD, exploiting the capability of redefining elementary operation semantics. + +A popular tool implemented with operator overloading in C++ is ADOL-C \citep{Walther2012}. ADOL-C requires the use of AD-enabled types for variables, and records arithmetic operations on variables in tape data structures, which can subsequently be ``played back'' during reverse mode AD computations. The Mxyzptlk package \citep{Michelotti1990} is another example for C++ capable of computing arbitrary-order partial derivatives via forward propagation. The FADBAD++ library \citep{Bendtsen1996} implements AD for C++ using templates and operator overloading. For Python, the \emph{ad} package\footnote{\url{http://pythonhosted.org/ad/}} uses operator overloading to compute first- and second-order derivatives, while the newer autograd package\footnote{\url{https://github.com/HIPS/autograd}} provides forward and reverse mode AD with support for higher-order derivatives. + +For functional languages, examples include R6RS-AD\footnote{\url{https://github.com/NUIM-BCL/R6RS-AD}} and the AD routines within the Scmutils library\footnote{\url{http://groups.csail.mit.edu/mac/users/gjs/6946/refman.txt}} for Scheme, the \emph{ad} library\footnote{\url{http://hackage.haskell.org/package/ad}} for Haskell, and DiffSharp\footnote{\url{http://diffsharp.github.io}} for F\# and C\#. + +%\subsection{Parallelization} +% Discuss advantages of AD for parallelization. +%Parallel implementation of Hessian calculations, automated generation of parallel code \citep{Bucker2008} +%\citet{Bischof2008} give an example of parallel reverse mode AD for a plasma simulation code + +\section{Conclusions} +\label{SectionConclusions} +Backpropagation and gradient-based optimization are behind virtually all recent successes in machine learning, yielding state-of-the-art results in computer vision, speech recognition and synthesis, and machine translation. We expect these techniques to remain at the core of machine learning for the foreseeable future. Research in the field involves a rapid prototyping and development cycle for testing new models and ideas, using a collection of increasingly higher-quality machine learning frameworks. These frameworks are in the process of transition from coarse-grained (module level) backpropagation towards fine-grained, general-purpose AD, allowing models to be implemented as regular programs in general-purpose programming languages with differentiation as an integral part of the infrastructure. We strongly believe that general-purpose AD is the future of gradient-based machine learning and we expect it to become an indispensable tool in the machine learning toolbox. + +It is an exciting time for working at the intersection of AD and machine learning, and there are many opportunities for bringing advanced techniques and expertise from AD literature to bear on machine learning problems. Techniques that have been developed by the AD community such as tape reduction and elimination \citep{naumann2004optimal}, fixed-point iterations \citep{christianson1994reverse}, utilizing sparsity by matrix coloring \citep{Gebremedhin2009,gebremedhin2013colpack}, and reverse AD checkpointing \citep{Dauvergne2006} are just a few examples that can find potential use in machine learning for increasing performance, improving convergence of optimization, using hardware more efficiently, and even enabling new types of machine learning models to be implemented. Similarly, exciting new AD modes like direct propagation of the inverse Jacobian \citep{srinivasan-todorov-2015a} have emerged from the machine learning community, but have yet to be examined and formalized by the AD community. + +An important direction for future work is to make use of nested AD techniques in machine learning, allowing differentiation to be nested arbitrarily deep with referential transparency \citep{Siskind2008b,pearlmutter2008reverse}. Nested AD is highly relevant in hyperparameter optimization as it can effortlessly provide exact hypergradients, that is, derivatives of a training objective with respect to the hyperparameters of an optimization routine \citep{Maclaurin2015,baydin2017online}. Potential applications include Bayesian model selection \citep{Rasmussen2006} and gradient-based tuning of Hamiltonian Monte Carlo step sizes and mass matrices \citep{Salimans2014}. Besides hyperparameters, models internally using higher-order derivatives constitute a straightforward usage case for nested AD. The Riemannian manifold Langevin and Hamiltonian Monte Carlo methods \citep{Girolami2011} use higher-order derivative information to more closely track the information geometry of the sampled distribution for faster convergence and exploration. In neural networks, it is very natural to use nested derivatives in defining objective functions that take input transformations into account, such as the Tangent Prop method \citep{Simard1998} for imposing invariance under a set of chosen transformations. + +% Acknowledgements should go at the end, before appendices and references + +\acks{We thank the anonymous reviewers whose comments helped improve this manuscript. This work was supported, in part, by Science Foundation Ireland grant 09/IN.1/I2637, by the Army Research Laboratory, accomplished under Cooperative Agreement Number W911NF-10-2-0060, by the National Science Foundation under Grants 1522954-IIS and 1734938-IIS, and by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior/Interior Business Center (DOI/IBC) contract number D17PC00341. Any opinions, findings, views, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views, official policies, or endorsements, either expressed or implied, of the sponsors. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes, notwithstanding any copyright notation herein. +} + +% Manual newpage inserted to improve layout of sample file - not +% needed in general before appendices/bibliography. +% +% \newpage +% +% \appendix +% \section*{Appendix A.} +% \label{app:theorem} + +% Note: in this sample, the section number is hard-coded in. Following +% proper LaTeX conventions, it should properly be coded as a reference: + +%In this appendix we prove the following theorem from +%Section~\ref{sec:textree-generalization}: + +\vskip 0.2in +\bibliography{17-468} + +\end{document} diff --git a/doc/Articles/AutodiffArticle b/doc/Articles/Autodiff/adiff.tar similarity index 100% rename from doc/Articles/AutodiffArticle rename to doc/Articles/Autodiff/adiff.tar diff --git a/doc/Articles/Autodiff/figures/approx-error/approx-error.tex b/doc/Articles/Autodiff/figures/approx-error/approx-error.tex new file mode 100644 index 000000000..3dec348fd --- /dev/null +++ b/doc/Articles/Autodiff/figures/approx-error/approx-error.tex @@ -0,0 +1,2189 @@ +% Created by tikzDevice version 0.8.1 on 2015-04-18 01:33:45 +% !TEX encoding = UTF-8 Unicode +\begin{tikzpicture}[x=1pt,y=1pt] +\definecolor{fillColor}{RGB}{255,255,255} +\path[use as bounding box,fill=fillColor,fill opacity=0.00] (0,0) rectangle (325.21,216.81); +\begin{scope} +\path[clip] ( 48.00, 48.00) rectangle (277.21,204.81); +\definecolor{drawColor}{RGB}{0,0,0} + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 56.49,198.53) -- + ( 56.70,198.53) -- + ( 56.91,198.53) -- + ( 57.13,198.53) -- + ( 57.34,198.53) -- + ( 57.55,198.53) -- + ( 57.76,198.53) -- + ( 57.98,198.53) -- + ( 58.19,198.53) -- + ( 58.40,198.53) -- + ( 58.61,197.02) -- + ( 58.83,196.62) -- + ( 59.04,196.19) -- + ( 59.25,195.74) -- + ( 59.46,195.26) -- + ( 59.68,194.74) -- + ( 59.89,194.19) -- + ( 60.10,193.58) -- + ( 60.31,192.92) -- + ( 60.53,192.17) -- + ( 60.74,191.33) -- + ( 60.95,190.34) -- + ( 61.16,189.15) -- + ( 61.38,187.63) -- + ( 61.59,185.50) -- + ( 61.80,181.82) -- + ( 62.01,170.59) -- + ( 62.23,182.54) -- + ( 62.44,185.61) -- + ( 62.65,187.42) -- + ( 62.86,188.70) -- + ( 63.08,189.68) -- + ( 63.29,190.47) -- + ( 63.50,191.13) -- + ( 63.71,191.68) -- + ( 63.93,192.17) -- + ( 64.14,192.60) -- + ( 64.35,192.98) -- + ( 64.56,193.32) -- + ( 64.77,193.63) -- + ( 64.99,193.90) -- + ( 65.20,194.16) -- + ( 65.41,170.80) -- + ( 65.62,181.65) -- + ( 65.84,185.17) -- + ( 66.05,187.14) -- + ( 66.26,188.49) -- + ( 66.47,189.51) -- + ( 66.69,190.33) -- + ( 66.90,191.01) -- + ( 67.11,191.59) -- + ( 67.32,192.08) -- + ( 67.54,192.52) -- + ( 67.75,192.91) -- + ( 67.96,193.26) -- + ( 68.17,193.57) -- + ( 68.39,193.85) -- + ( 68.60,190.59) -- + ( 68.81,189.45) -- + ( 69.02,188.03) -- + ( 69.24,186.09) -- + ( 69.45,182.95) -- + ( 69.66,172.81) -- + ( 69.87,181.38) -- + ( 70.09,185.05) -- + ( 70.30,187.06) -- + ( 70.51,188.45) -- + ( 70.72,186.68) -- + ( 70.94,184.00) -- + ( 71.15,177.95) -- + ( 71.36,179.58) -- + ( 71.57,184.31) -- + ( 71.79,186.60) -- + ( 72.00,188.10) -- + ( 72.21,184.17) -- + ( 72.42,186.52) -- + ( 72.64,188.04) -- + ( 72.85,189.17) -- + ( 73.06,190.05) -- + ( 73.27,188.11) -- + ( 73.49,186.20) -- + ( 73.70,183.16) -- + ( 73.91,174.22) -- + ( 74.12,181.10) -- + ( 74.34,185.87) -- + ( 74.55,182.55) -- + ( 74.76,168.10) -- + ( 74.97,181.85) -- + ( 75.18,184.17) -- + ( 75.40,178.49) -- + ( 75.61,179.16) -- + ( 75.82,184.17) -- + ( 76.03,178.97) -- + ( 76.25,178.72) -- + ( 76.46,184.02) -- + ( 76.67,175.80) -- + ( 76.88,180.65) -- + ( 77.10,184.73) -- + ( 77.31,185.98) -- + ( 77.52,182.75) -- + ( 77.73,170.92) -- + ( 77.95,177.11) -- + ( 78.16,183.55) -- + ( 78.37,180.88) -- + ( 78.58,175.46) -- + ( 78.80,180.15) -- + ( 79.01,177.12) -- + ( 79.22,178.13) -- + ( 79.43,179.45) -- + ( 79.65,181.82) -- + ( 79.86,170.58) -- + ( 80.07,180.40) -- + ( 80.28,184.63) -- + ( 80.50,175.91) -- + ( 80.71,180.16) -- + ( 80.92,177.10) -- + ( 81.13,166.06) -- + ( 81.35,170.38) -- + ( 81.56,182.52) -- + ( 81.77,179.46) -- + ( 81.98,180.51) -- + ( 82.20,178.93) -- + ( 82.41,178.76) -- + ( 82.62,180.49) -- + ( 82.83,174.41) -- + ( 83.05,175.17) -- + ( 83.26,173.07) -- + ( 83.47,179.86) -- + ( 83.68,176.47) -- + ( 83.90,167.71) -- + ( 84.11,175.29) -- + ( 84.32,180.05) -- + ( 84.53,175.35) -- + ( 84.75,165.91) -- + ( 84.96,176.79) -- + ( 85.17,176.38) -- + ( 85.38,173.71) -- + ( 85.59,174.26) -- + ( 85.81,178.64) -- + ( 86.02,170.23) -- + ( 86.23,176.90) -- + ( 86.44,171.84) -- + ( 86.66,167.85) -- + ( 86.87,178.66) -- + ( 87.08,177.71) -- + ( 87.29,174.09) -- + ( 87.51,164.60) -- + ( 87.72,163.08) -- + ( 87.93,173.08) -- + ( 88.14,176.52) -- + ( 88.36,174.20) -- + ( 88.57,175.80) -- + ( 88.78,173.61) -- + ( 88.99,169.67) -- + ( 89.21,163.34) -- + ( 89.42,153.14) -- + ( 89.63,163.02) -- + ( 89.84,169.17) -- + ( 90.06,172.91) -- + ( 90.27,166.13) -- + ( 90.48,171.23) -- + ( 90.69,169.28) -- + ( 90.91,174.23) -- + ( 91.12,166.34) -- + ( 91.33,173.46) -- + ( 91.54,163.55) -- + ( 91.76,165.93) -- + ( 91.97,165.07) -- + ( 92.18,167.75) -- + ( 92.39,171.38) -- + ( 92.61,156.59) -- + ( 92.82,169.11) -- + ( 93.03,168.83) -- + ( 93.24,171.38) -- + ( 93.46,169.40) -- + ( 93.67,167.29) -- + ( 93.88,167.79) -- + ( 94.09,163.09) -- + ( 94.31,161.59) -- + ( 94.52,171.98) -- + ( 94.73,160.88) -- + ( 94.94,161.05) -- + ( 95.16,167.20) -- + ( 95.37,160.14) -- + ( 95.58,157.32) -- + ( 95.79,164.73) -- + ( 96.00,166.86) -- + ( 96.22,165.56) -- + ( 96.43,164.21) -- + ( 96.64,158.41) -- + ( 96.85,167.24) -- + ( 97.07,167.99) -- + ( 97.28,155.09) -- + ( 97.49,167.22) -- + ( 97.70,160.72) -- + ( 97.92,167.82) -- + ( 98.13,153.40) -- + ( 98.34,159.65) -- + ( 98.55,168.79) -- + ( 98.77,167.50) -- + ( 98.98,166.99) -- + ( 99.19,167.10) -- + ( 99.40,163.43) -- + ( 99.62,167.84) -- + ( 99.83,157.93) -- + (100.04,162.37) -- + (100.25,163.99) -- + (100.47,164.35) -- + (100.68,163.29) -- + (100.89,163.95) -- + (101.10,161.94) -- + (101.32,165.18) -- + (101.53,166.14) -- + (101.74,154.03) -- + (101.95,155.95) -- + (102.17,165.85) -- + (102.38,164.46) -- + (102.59,144.38) -- + (102.80,151.00) -- + (103.02,143.62) -- + (103.23,163.78) -- + (103.44,159.34) -- + (103.65,156.64) -- + (103.87,156.25) -- + (104.08,154.87) -- + (104.29,159.20) -- + (104.50,154.69) -- + (104.72,162.29) -- + (104.93,156.53) -- + (105.14,158.05) -- + (105.35,161.54) -- + (105.57,157.64) -- + (105.78,156.77) -- + (105.99,159.95) -- + (106.20,159.93) -- + (106.41,157.96) -- + (106.63,157.86) -- + (106.84,147.05) -- + (107.05,155.97) -- + (107.26,161.82) -- + (107.48,156.20) -- + (107.69,160.54) -- + (107.90,151.53) -- + (108.11,159.99) -- + (108.33,150.91) -- + (108.54,148.23) -- + (108.75,152.46) -- + (108.96,160.35) -- + (109.18,154.71) -- + (109.39,154.76) -- + (109.60,154.73) -- + (109.81,157.94) -- + (110.03,154.45) -- + (110.24,156.18) -- + (110.45,157.29) -- + (110.66,158.80) -- + (110.88,150.42) -- + (111.09,150.17) -- + (111.30,148.86) -- + (111.51,151.11) -- + (111.73,155.62) -- + (111.94,158.59) -- + (112.15,156.21) -- + (112.36,150.54) -- + (112.58,148.92) -- + (112.79,154.22) -- + (113.00,156.36) -- + (113.21,152.79) -- + (113.43,151.63) -- + (113.64,140.40) -- + (113.85,155.64) -- + (114.06,145.62) -- + (114.28,150.93) -- + (114.49,154.13) -- + (114.70,155.59) -- + (114.91,155.58) -- + (115.13,152.15) -- + (115.34,152.34) -- + (115.55,148.95) -- + (115.76,138.48) -- + (115.98,151.81) -- + (116.19,143.12) -- + (116.40,153.36) -- + (116.61,147.05) -- + (116.82,146.19) -- + (117.04,145.87) -- + (117.25,146.29) -- + (117.46,152.82) -- + (117.67,151.54) -- + (117.89,153.37) -- + (118.10,146.14) -- + (118.31,144.76) -- + (118.52,151.18) -- + (118.74,149.11) -- + (118.95,149.15) -- + (119.16,142.89) -- + (119.37,151.48) -- + (119.59,143.52) -- + (119.80,144.21) -- + (120.01,142.03) -- + (120.22,148.72) -- + (120.44,150.24) -- + (120.65,144.99) -- + (120.86,148.89) -- + (121.07,147.03) -- + (121.29,141.75) -- + (121.50,139.54) -- + (121.71,146.96) -- + (121.92,147.88) -- + (122.14,138.86) -- + (122.35,146.14) -- + (122.56,140.96) -- + (122.77,141.99) -- + (122.99,138.78) -- + (123.20,146.42) -- + (123.41,140.13) -- + (123.62,146.49) -- + (123.84,142.64) -- + (124.05,143.53) -- + (124.26,147.33) -- + (124.47,147.10) -- + (124.69,140.97) -- + (124.90,141.93) -- + (125.11,142.85) -- + (125.32,142.95) -- + (125.54,135.83) -- + (125.75,146.34) -- + (125.96,141.41) -- + (126.17,146.44) -- + (126.39,146.59) -- + (126.60,136.49) -- + (126.81,135.68) -- + (127.02,145.90) -- + (127.23,140.74) -- + (127.45,135.18) -- + (127.66,138.47) -- + (127.87,141.16) -- + (128.08,144.06) -- + (128.30,140.13) -- + (128.51,140.55) -- + (128.72,119.01) -- + (128.93,140.49) -- + (129.15,137.70) -- + (129.36,134.89) -- + (129.57,131.77) -- + (129.78,142.82) -- + (130.00,141.57) -- + (130.21,142.23) -- + (130.42,135.75) -- + (130.63,141.36) -- + (130.85,143.52) -- + (131.06,143.28) -- + (131.27,138.78) -- + (131.48,136.05) -- + (131.70,124.23) -- + (131.91,140.32) -- + (132.12,141.15) -- + (132.33,138.86) -- + (132.55,136.70) -- + (132.76,136.72) -- + (132.97,137.85) -- + (133.18,140.37) -- + (133.40,134.90) -- + (133.61,136.29) -- + (133.82,136.73) -- + (134.03,138.83) -- + (134.25,141.96) -- + (134.46,134.92) -- + (134.67,138.38) -- + (134.88,126.82) -- + (135.10,134.02) -- + (135.31,131.44) -- + (135.52,138.83) -- + (135.73,125.96) -- + (135.95,138.36) -- + (136.16,129.42) -- + (136.37,136.91) -- + (136.58,127.75) -- + (136.79,126.23) -- + (137.01,136.81) -- + (137.22,130.85) -- + (137.43,129.91) -- + (137.64,133.30) -- + (137.86,136.14) -- + (138.07,130.05) -- + (138.28,132.85) -- + (138.49,129.33) -- + (138.71,126.72) -- + (138.92,133.09) -- + (139.13,137.27) -- + (139.34,133.38) -- + (139.56,135.56) -- + (139.77,126.70) -- + (139.98,132.28) -- + (140.19,126.01) -- + (140.41,127.92) -- + (140.62,131.44) -- + (140.83,131.22) -- + (141.04,129.73) -- + (141.26,130.22) -- + (141.47,134.27) -- + (141.68,123.89) -- + (141.89,134.70) -- + (142.11,133.57) -- + (142.32,133.67) -- + (142.53,129.45) -- + (142.74,130.76) -- + (142.96,128.44) -- + (143.17,116.31) -- + (143.38,133.47) -- + (143.59,131.21) -- + (143.81,122.91) -- + (144.02,124.66) -- + (144.23,130.84) -- + (144.44,131.72) -- + (144.66,124.11) -- + (144.87,126.24) -- + (145.08,127.60) -- + (145.29,130.31) -- + (145.51,133.26) -- + (145.72,129.24) -- + (145.93,128.36) -- + (146.14,128.26) -- + (146.36,124.90) -- + (146.57,129.31) -- + (146.78,121.12) -- + (146.99,127.78) -- + (147.20,129.79) -- + (147.42,123.09) -- + (147.63,128.52) -- + (147.84,118.16) -- + (148.05,114.64) -- + (148.27,128.06) -- + (148.48,128.17) -- + (148.69,122.77) -- + (148.90,126.70) -- + (149.12,125.57) -- + (149.33,128.36) -- + (149.54,120.82) -- + (149.75,126.42) -- + (149.97,128.15) -- + (150.18,121.66) -- + (150.39,126.36) -- + (150.60,120.26) -- + (150.82,114.23) -- + (151.03,126.66) -- + (151.24,118.20) -- + (151.45,121.76) -- + (151.67,119.24) -- + (151.88,115.04) -- + (152.09,118.47) -- + (152.30,123.20) -- + (152.52,118.29) -- + (152.73,117.92) -- + (152.94,124.38) -- + (153.15,116.04) -- + (153.37,108.10) -- + (153.58,125.57) -- + (153.79,115.40) -- + (154.00,114.20) -- + (154.22,113.87) -- + (154.43,100.39) -- + (154.64,122.11) -- + (154.85,118.63) -- + (155.07,123.71) -- + (155.28, 94.92) -- + (155.49,113.78) -- + (155.70,123.41) -- + (155.92,115.89) -- + (156.13,117.82) -- + (156.34,122.75) -- + (156.55,117.34) -- + (156.77,109.81) -- + (156.98,121.32) -- + (157.19,122.26) -- + (157.40,119.88) -- + (157.61,118.06) -- + (157.83,123.42) -- + (158.04,116.26) -- + (158.25,112.68) -- + (158.46,115.73) -- + (158.68,119.85) -- + (158.89,116.04) -- + (159.10,117.34) -- + (159.31,108.92) -- + (159.53,100.17) -- + (159.74,120.23) -- + (159.95,118.00) -- + (160.16,119.84) -- + (160.38,118.62) -- + (160.59,110.23) -- + (160.80,117.91) -- + (161.01,118.93) -- + (161.23,116.17) -- + (161.44,107.43) -- + (161.65,112.46) -- + (161.86,111.57) -- + (162.08,115.90) -- + (162.29,115.08) -- + (162.50, 99.68) -- + (162.71,118.69) -- + (162.93,115.24) -- + (163.14,105.78) -- + (163.35,115.67) -- + (163.56,109.42) -- + (163.78,107.04) -- + (163.99,112.08) -- + (164.20,106.40) -- + (164.41,116.63) -- + (164.63,114.54) -- + (164.84,110.43) -- + (165.05,111.22) -- + (165.26, 80.52) -- + (165.48,113.00) -- + (165.69,112.20) -- + (165.90, 91.06) -- + (166.11,115.48) -- + (166.33, 99.57) -- + (166.54,107.63) -- + (166.75,107.97) -- + (166.96,115.27) -- + (167.18,105.03) -- + (167.39,110.73) -- + (167.60,106.80) -- + (167.81,102.32) -- + (168.02,104.99) -- + (168.24,101.91) -- + (168.45,110.31) -- + (168.66,105.03) -- + (168.87,109.14) -- + (169.09,111.77) -- + (169.30,110.41) -- + (169.51,109.44) -- + (169.72,111.63) -- + (169.94,108.75) -- + (170.15,111.65) -- + (170.36,103.43) -- + (170.57,105.82) -- + (170.79, 87.49) -- + (171.00,108.83) -- + (171.21,109.03) -- + (171.42,111.11) -- + (171.64,107.82) -- + (171.85,108.28) -- + (172.06, 91.72) -- + (172.27,101.96) -- + (172.49,106.62) -- + (172.70,110.04) -- + (172.91,105.97) -- + (173.12,110.52) -- + (173.34,109.25) -- + (173.55, 97.72) -- + (173.76, 96.77) -- + (173.97,108.15) -- + (174.19,106.73) -- + (174.40,104.52) -- + (174.61,112.22) -- + (174.82,110.05) -- + (175.04,102.72) -- + (175.25,106.11) -- + (175.46,109.89) -- + (175.67,106.90) -- + (175.89,110.24) -- + (176.10,110.07) -- + (176.31,105.54) -- + (176.52,110.97) -- + (176.74,108.56) -- + (176.95,111.67) -- + (177.16,109.94) -- + (177.37,110.12) -- + (177.59,108.26) -- + (177.80,110.49) -- + (178.01,111.10) -- + (178.22,110.60) -- + (178.43,111.01) -- + (178.65,112.35) -- + (178.86,111.42) -- + (179.07,110.55) -- + (179.28,112.80) -- + (179.50,111.40) -- + (179.71,112.62) -- + (179.92,112.26) -- + (180.13,112.71) -- + (180.35,112.39) -- + (180.56,112.50) -- + (180.77,112.67) -- + (180.98,113.70) -- + (181.20,113.46) -- + (181.41,113.38) -- + (181.62,114.35) -- + (181.83,113.81) -- + (182.05,114.64) -- + (182.26,114.62) -- + (182.47,114.59) -- + (182.68,114.31) -- + (182.90,114.73) -- + (183.11,114.85) -- + (183.32,115.13) -- + (183.53,115.36) -- + (183.75,115.52) -- + (183.96,115.58) -- + (184.17,115.72) -- + (184.38,116.01) -- + (184.60,116.28) -- + (184.81,116.27) -- + (185.02,116.51) -- + (185.23,116.78) -- + (185.45,116.84) -- + (185.66,116.96) -- + (185.87,117.19) -- + (186.08,117.25) -- + (186.30,117.60) -- + (186.51,117.59) -- + (186.72,117.90) -- + (186.93,117.94) -- + (187.15,118.18) -- + (187.36,118.31) -- + (187.57,118.48) -- + (187.78,118.63) -- + (188.00,118.82) -- + (188.21,118.98) -- + (188.42,119.14) -- + (188.63,119.34) -- + (188.84,119.47) -- + (189.06,119.67) -- + (189.27,119.83) -- + (189.48,119.99) -- + (189.69,120.18) -- + (189.91,120.34) -- + (190.12,120.51) -- + (190.33,120.65) -- + (190.54,120.86) -- + (190.76,121.01) -- + (190.97,121.18) -- + (191.18,121.35) -- + (191.39,121.49) -- + (191.61,121.66) -- + (191.82,121.84) -- + (192.03,122.02) -- + (192.24,122.19) -- + (192.46,122.36) -- + (192.67,122.53) -- + (192.88,122.69) -- + (193.09,122.85) -- + (193.31,123.03) -- + (193.52,123.19) -- + (193.73,123.37) -- + (193.94,123.52) -- + (194.16,123.69) -- + (194.37,123.86) -- + (194.58,124.03) -- + (194.79,124.20) -- + (195.01,124.36) -- + (195.22,124.54) -- + (195.43,124.70) -- + (195.64,124.87) -- + (195.86,125.03) -- + (196.07,125.21) -- + (196.28,125.38) -- + (196.49,125.54) -- + (196.71,125.71) -- + (196.92,125.87) -- + (197.13,126.04) -- + (197.34,126.21) -- + (197.56,126.38) -- + (197.77,126.55) -- + (197.98,126.71) -- + (198.19,126.88) -- + (198.41,127.05) -- + (198.62,127.22) -- + (198.83,127.38) -- + (199.04,127.55) -- + (199.25,127.72) -- + (199.47,127.89) -- + (199.68,128.06) -- + (199.89,128.22) -- + (200.10,128.39) -- + (200.32,128.56) -- + (200.53,128.72) -- + (200.74,128.89) -- + (200.95,129.06) -- + (201.17,129.23) -- + (201.38,129.40) -- + (201.59,129.56) -- + (201.80,129.73) -- + (202.02,129.90) -- + (202.23,130.07) -- + (202.44,130.24) -- + (202.65,130.40) -- + (202.87,130.57) -- + (203.08,130.74) -- + (203.29,130.91) -- + (203.50,131.07) -- + (203.72,131.24) -- + (203.93,131.41) -- + (204.14,131.58) -- + (204.35,131.74) -- + (204.57,131.91) -- + (204.78,132.08) -- + (204.99,132.25) -- + (205.20,132.42) -- + (205.42,132.58) -- + (205.63,132.75) -- + (205.84,132.92) -- + (206.05,133.09) -- + (206.27,133.25) -- + (206.48,133.42) -- + (206.69,133.59) -- + (206.90,133.76) -- + (207.12,133.92) -- + (207.33,134.09) -- + (207.54,134.26) -- + (207.75,134.43) -- + (207.97,134.60) -- + (208.18,134.76) -- + (208.39,134.93) -- + (208.60,135.10) -- + (208.82,135.27) -- + (209.03,135.43) -- + (209.24,135.60) -- + (209.45,135.77) -- + (209.66,135.94) -- + (209.88,136.10) -- + (210.09,136.27) -- + (210.30,136.44) -- + (210.51,136.61) -- + (210.73,136.78) -- + (210.94,136.94) -- + (211.15,137.11) -- + (211.36,137.28) -- + (211.58,137.45) -- + (211.79,137.61) -- + (212.00,137.78) -- + (212.21,137.95) -- + (212.43,138.12) -- + (212.64,138.28) -- + (212.85,138.45) -- + (213.06,138.62) -- + (213.28,138.79) -- + (213.49,138.96) -- + (213.70,139.12) -- + (213.91,139.29) -- + (214.13,139.46) -- + (214.34,139.63) -- + (214.55,139.79) -- + (214.76,139.96) -- + (214.98,140.13) -- + (215.19,140.30) -- + (215.40,140.46) -- + (215.61,140.63) -- + (215.83,140.80) -- + (216.04,140.97) -- + (216.25,141.14) -- + (216.46,141.30) -- + (216.68,141.47) -- + (216.89,141.64) -- + (217.10,141.81) -- + (217.31,141.97) -- + (217.53,142.14) -- + (217.74,142.31) -- + (217.95,142.48) -- + (218.16,142.64) -- + (218.38,142.81) -- + (218.59,142.98) -- + (218.80,143.15) -- + (219.01,143.32) -- + (219.23,143.48) -- + (219.44,143.65) -- + (219.65,143.82) -- + (219.86,143.99) -- + (220.07,144.15) -- + (220.29,144.32) -- + (220.50,144.49) -- + (220.71,144.66) -- + (220.92,144.82) -- + (221.14,144.99) -- + (221.35,145.16) -- + (221.56,145.33) -- + (221.77,145.50) -- + (221.99,145.66) -- + (222.20,145.83) -- + (222.41,146.00) -- + (222.62,146.17) -- + (222.84,146.33) -- + (223.05,146.50) -- + (223.26,146.67) -- + (223.47,146.84) -- + (223.69,147.00) -- + (223.90,147.17) -- + (224.11,147.34) -- + (224.32,147.51) -- + (224.54,147.68) -- + (224.75,147.84) -- + (224.96,148.01) -- + (225.17,148.18) -- + (225.39,148.35) -- + (225.60,148.51) -- + (225.81,148.68) -- + (226.02,148.85) -- + (226.24,149.02) -- + (226.45,149.18) -- + (226.66,149.35) -- + (226.87,149.52) -- + (227.09,149.69) -- + (227.30,149.85) -- + (227.51,150.02) -- + (227.72,150.19) -- + (227.94,150.36) -- + (228.15,150.53) -- + (228.36,150.69) -- + (228.57,150.86) -- + (228.79,151.03) -- + (229.00,151.20) -- + (229.21,151.36) -- + (229.42,151.53) -- + (229.64,151.70) -- + (229.85,151.87) -- + (230.06,152.03) -- + (230.27,152.20) -- + (230.48,152.37) -- + (230.70,152.54) -- + (230.91,152.70) -- + (231.12,152.87) -- + (231.33,153.04) -- + (231.55,153.21) -- + (231.76,153.38) -- + (231.97,153.54) -- + (232.18,153.71) -- + (232.40,153.88) -- + (232.61,154.05) -- + (232.82,154.21) -- + (233.03,154.38) -- + (233.25,154.55) -- + (233.46,154.72) -- + (233.67,154.88) -- + (233.88,155.05) -- + (234.10,155.22) -- + (234.31,155.39) -- + (234.52,155.55) -- + (234.73,155.72) -- + (234.95,155.89) -- + (235.16,156.06) -- + (235.37,156.22) -- + (235.58,156.39) -- + (235.80,156.56) -- + (236.01,156.73) -- + (236.22,156.89) -- + (236.43,157.06) -- + (236.65,157.23) -- + (236.86,157.40) -- + (237.07,157.56) -- + (237.28,157.73) -- + (237.50,157.90) -- + (237.71,158.07) -- + (237.92,158.23) -- + (238.13,158.40) -- + (238.35,158.57) -- + (238.56,158.74) -- + (238.77,158.90) -- + (238.98,159.07) -- + (239.20,159.24) -- + (239.41,159.41) -- + (239.62,159.57) -- + (239.83,159.74) -- + (240.04,159.91) -- + (240.26,160.08) -- + (240.47,160.24) -- + (240.68,160.41) -- + (240.89,160.58) -- + (241.11,160.75) -- + (241.32,160.91) -- + (241.53,161.08) -- + (241.74,161.25) -- + (241.96,161.41) -- + (242.17,161.58) -- + (242.38,161.75) -- + (242.59,161.92) -- + (242.81,162.08) -- + (243.02,162.25) -- + (243.23,162.42) -- + (243.44,162.59) -- + (243.66,162.75) -- + (243.87,162.92) -- + (244.08,163.09) -- + (244.29,163.25) -- + (244.51,163.42) -- + (244.72,163.59) -- + (244.93,163.76) -- + (245.14,163.92) -- + (245.36,164.09) -- + (245.57,164.26) -- + (245.78,164.42) -- + (245.99,164.59) -- + (246.21,164.76) -- + (246.42,164.92) -- + (246.63,165.09) -- + (246.84,165.26) -- + (247.06,165.42) -- + (247.27,165.59) -- + (247.48,165.76) -- + (247.69,165.92) -- + (247.91,166.09) -- + (248.12,166.26) -- + (248.33,166.42) -- + (248.54,166.59) -- + (248.76,166.76) -- + (248.97,166.92) -- + (249.18,167.09) -- + (249.39,167.26) -- + (249.61,167.42) -- + (249.82,167.59) -- + (250.03,167.75) -- + (250.24,167.92) -- + (250.45,168.09) -- + (250.67,168.25) -- + (250.88,168.42) -- + (251.09,168.58) -- + (251.30,168.75) -- + (251.52,168.92) -- + (251.73,169.08) -- + (251.94,169.25) -- + (252.15,169.41) -- + (252.37,169.58) -- + (252.58,169.74) -- + (252.79,169.91) -- + (253.00,170.07) -- + (253.22,170.24) -- + (253.43,170.40) -- + (253.64,170.57) -- + (253.85,170.73) -- + (254.07,170.90) -- + (254.28,171.06) -- + (254.49,171.23) -- + (254.70,171.39) -- + (254.92,171.56) -- + (255.13,171.72) -- + (255.34,171.88) -- + (255.55,172.05) -- + (255.77,172.21) -- + (255.98,172.37) -- + (256.19,172.54) -- + (256.40,172.70) -- + (256.62,172.86) -- + (256.83,173.03) -- + (257.04,173.19) -- + (257.25,173.35) -- + (257.47,173.52) -- + (257.68,173.68) -- + (257.89,173.84) -- + (258.10,174.00) -- + (258.32,174.16) -- + (258.53,174.32) -- + (258.74,174.49) -- + (258.95,174.65) -- + (259.17,174.81) -- + (259.38,174.97) -- + (259.59,175.13) -- + (259.80,175.29) -- + (260.02,175.45) -- + (260.23,175.61) -- + (260.44,175.77) -- + (260.65,175.92) -- + (260.86,176.08) -- + (261.08,176.24) -- + (261.29,176.40) -- + (261.50,176.55) -- + (261.71,176.71) -- + (261.93,176.87) -- + (262.14,177.02) -- + (262.35,177.18) -- + (262.56,177.33) -- + (262.78,177.49) -- + (262.99,177.64) -- + (263.20,177.80) -- + (263.41,177.95) -- + (263.63,178.10) -- + (263.84,178.25) -- + (264.05,178.41) -- + (264.26,178.56) -- + (264.48,178.71) -- + (264.69,178.86) -- + (264.90,179.00) -- + (265.11,179.15) -- + (265.33,179.30) -- + (265.54,179.45) -- + (265.75,179.59) -- + (265.96,179.74) -- + (266.18,179.88) -- + (266.39,180.02) -- + (266.60,180.16) -- + (266.81,180.30) -- + (267.03,180.44) -- + (267.24,180.58) -- + (267.45,180.72) -- + (267.66,180.86) -- + (267.88,180.99) -- + (268.09,181.12) -- + (268.30,181.25) -- + (268.51,181.39) -- + (268.73,181.51); +\end{scope} +\begin{scope} +\path[clip] ( 0.00, 0.00) rectangle (325.21,216.81); +\definecolor{drawColor}{RGB}{0,0,0} + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00, 48.00) -- + (277.21, 48.00) -- + (277.21,204.81) -- + ( 48.00,204.81) -- + ( 48.00, 48.00); +\end{scope} +\begin{scope} +\path[clip] ( 0.00, 0.00) rectangle (325.21,216.81); +\definecolor{drawColor}{RGB}{0,0,0} + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at (162.61, 2.40) {$h$}; + +\node[text=drawColor,rotate= 90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 9.60,126.41) {Error}; +\end{scope} +\begin{scope} +\path[clip] ( 48.00, 48.00) rectangle (277.21,204.81); +\definecolor{drawColor}{RGB}{0,0,0} + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 4pt off 4pt ,line join=round,line cap=round] ( 56.49,198.53) -- + ( 56.70,198.53) -- + ( 56.91,198.53) -- + ( 57.13,198.53) -- + ( 57.34,198.53) -- + ( 57.55,198.53) -- + ( 57.76,198.53) -- + ( 57.98,198.53) -- + ( 58.19,198.53) -- + ( 58.40,198.53) -- + ( 58.61,195.32) -- + ( 58.83,194.81) -- + ( 59.04,194.26) -- + ( 59.25,193.67) -- + ( 59.46,193.01) -- + ( 59.68,192.28) -- + ( 59.89,191.45) -- + ( 60.10,190.49) -- + ( 60.31,189.33) -- + ( 60.53,187.87) -- + ( 60.74,185.85) -- + ( 60.95,182.51) -- + ( 61.16,167.33) -- + ( 61.38,181.90) -- + ( 61.59,185.29) -- + ( 61.80,187.21) -- + ( 62.01,188.55) -- + ( 62.23,189.56) -- + ( 62.44,190.37) -- + ( 62.65,191.04) -- + ( 62.86,191.61) -- + ( 63.08,192.11) -- + ( 63.29,192.54) -- + ( 63.50,192.93) -- + ( 63.71,193.27) -- + ( 63.93,193.58) -- + ( 64.14,193.87) -- + ( 64.35,194.13) -- + ( 64.56,194.36) -- + ( 64.77,194.58) -- + ( 64.99,194.79) -- + ( 65.20,194.97) -- + ( 65.41,189.21) -- + ( 65.62,187.72) -- + ( 65.84,185.63) -- + ( 66.05,182.08) -- + ( 66.26,166.90) -- + ( 66.47,182.32) -- + ( 66.69,185.50) -- + ( 66.90,187.35) -- + ( 67.11,188.65) -- + ( 67.32,189.64) -- + ( 67.54,190.43) -- + ( 67.75,191.10) -- + ( 67.96,191.66) -- + ( 68.17,192.15) -- + ( 68.39,192.58) -- + ( 68.60,191.55) -- + ( 68.81,190.60) -- + ( 69.02,189.47) -- + ( 69.24,188.05) -- + ( 69.45,186.12) -- + ( 69.66,183.02) -- + ( 69.87,173.29) -- + ( 70.09,181.30) -- + ( 70.30,185.01) -- + ( 70.51,189.53) -- + ( 70.72,188.13) -- + ( 70.94,186.23) -- + ( 71.15,183.21) -- + ( 71.36,174.54) -- + ( 71.57,181.03) -- + ( 71.79,184.89) -- + ( 72.00,186.96) -- + ( 72.21,179.37) -- + ( 72.42,178.27) -- + ( 72.64,183.88) -- + ( 72.85,186.34) -- + ( 73.06,187.92) -- + ( 73.27,187.20) -- + ( 73.49,184.84) -- + ( 73.70,180.33) -- + ( 73.91,176.78) -- + ( 74.12,183.47) -- + ( 74.34,186.87) -- + ( 74.55,184.31) -- + ( 74.76,178.92) -- + ( 74.97,178.77) -- + ( 75.18,184.17) -- + ( 75.40,178.49) -- + ( 75.61,179.16) -- + ( 75.82,184.17) -- + ( 76.03,178.97) -- + ( 76.25,178.72) -- + ( 76.46,184.02) -- + ( 76.67,183.33) -- + ( 76.88,175.20) -- + ( 77.10,180.84) -- + ( 77.31,185.98) -- + ( 77.52,182.75) -- + ( 77.73,170.92) -- + ( 77.95,169.95) -- + ( 78.16,182.49) -- + ( 78.37,178.93) -- + ( 78.58,178.76) -- + ( 78.80,180.15) -- + ( 79.01,177.12) -- + ( 79.22,182.62) -- + ( 79.43,169.21) -- + ( 79.65,181.82) -- + ( 79.86,170.58) -- + ( 80.07,178.83) -- + ( 80.28,184.06) -- + ( 80.50,178.34) -- + ( 80.71,182.21) -- + ( 80.92,162.28) -- + ( 81.13,172.11) -- + ( 81.35,174.92) -- + ( 81.56,180.92) -- + ( 81.77,178.03) -- + ( 81.98,181.39) -- + ( 82.20,177.43) -- + ( 82.41,179.90) -- + ( 82.62,178.33) -- + ( 82.83,174.41) -- + ( 83.05,171.79) -- + ( 83.26,173.07) -- + ( 83.47,178.91) -- + ( 83.68,170.77) -- + ( 83.90,173.14) -- + ( 84.11,175.29) -- + ( 84.32,179.27) -- + ( 84.53,179.03) -- + ( 84.75,172.79) -- + ( 84.96,175.31) -- + ( 85.17,172.47) -- + ( 85.38,175.56) -- + ( 85.59,174.26) -- + ( 85.81,179.35) -- + ( 86.02,160.85) -- + ( 86.23,175.77) -- + ( 86.44,167.72) -- + ( 86.66,167.85) -- + ( 86.87,178.66) -- + ( 87.08,175.93) -- + ( 87.29,172.26) -- + ( 87.51,164.60) -- + ( 87.72,166.02) -- + ( 87.93,167.28) -- + ( 88.14,175.64) -- + ( 88.36,175.32) -- + ( 88.57,174.85) -- + ( 88.78,173.61) -- + ( 88.99,165.32) -- + ( 89.21,163.34) -- + ( 89.42,166.49) -- + ( 89.63,163.02) -- + ( 89.84,165.06) -- + ( 90.06,165.85) -- + ( 90.27,166.13) -- + ( 90.48,171.23) -- + ( 90.69,169.28) -- + ( 90.91,174.99) -- + ( 91.12,166.34) -- + ( 91.33,173.46) -- + ( 91.54,158.86) -- + ( 91.76,165.93) -- + ( 91.97,165.07) -- + ( 92.18,153.04) -- + ( 92.39,170.05) -- + ( 92.61,156.59) -- + ( 92.82,169.11) -- + ( 93.03,170.34) -- + ( 93.24,172.30) -- + ( 93.46,171.70) -- + ( 93.67,164.36) -- + ( 93.88,160.27) -- + ( 94.09,166.46) -- + ( 94.31,169.36) -- + ( 94.52,171.19) -- + ( 94.73,165.18) -- + ( 94.94,165.15) -- + ( 95.16,169.84) -- + ( 95.37,164.58) -- + ( 95.58,157.32) -- + ( 95.79,160.94) -- + ( 96.00,168.27) -- + ( 96.22,162.87) -- + ( 96.43,166.32) -- + ( 96.64,156.93) -- + ( 96.85,163.43) -- + ( 97.07,166.75) -- + ( 97.28,158.70) -- + ( 97.49,160.73) -- + ( 97.70,167.10) -- + ( 97.92,165.29) -- + ( 98.13,158.45) -- + ( 98.34,159.65) -- + ( 98.55,168.79) -- + ( 98.77,165.20) -- + ( 98.98,167.88) -- + ( 99.19,166.07) -- + ( 99.40,161.01) -- + ( 99.62,167.04) -- + ( 99.83,159.14) -- + (100.04,159.53) -- + (100.25,163.99) -- + (100.47,160.50) -- + (100.68,161.35) -- + (100.89,155.27) -- + (101.10,159.42) -- + (101.32,166.07) -- + (101.53,165.29) -- + (101.74,153.35) -- + (101.95,148.49) -- + (102.17,165.85) -- + (102.38,164.46) -- + (102.59,156.79) -- + (102.80,154.21) -- + (103.02,161.51) -- + (103.23,161.36) -- + (103.44,155.18) -- + (103.65,156.64) -- + (103.87,161.02) -- + (104.08,154.87) -- + (104.29,159.20) -- + (104.50,139.15) -- + (104.72,163.22) -- + (104.93,151.14) -- + (105.14,154.97) -- + (105.35,160.34) -- + (105.57,160.89) -- + (105.78,156.77) -- + (105.99,156.05) -- + (106.20,152.00) -- + (106.41,150.69) -- + (106.63,155.51) -- + (106.84,151.37) -- + (107.05,159.47) -- + (107.26,161.01) -- + (107.48,156.20) -- + (107.69,159.54) -- + (107.90,155.87) -- + (108.11,157.60) -- + (108.33,150.91) -- + (108.54,148.23) -- + (108.75,157.19) -- + (108.96,158.53) -- + (109.18,151.50) -- + (109.39,154.76) -- + (109.60,156.53) -- + (109.81,158.92) -- + (110.03,143.72) -- + (110.24,154.44) -- + (110.45,156.01) -- + (110.66,157.94) -- + (110.88,155.38) -- + (111.09,153.30) -- + (111.30,154.64) -- + (111.51,156.58) -- + (111.73,155.62) -- + (111.94,157.87) -- + (112.15,156.21) -- + (112.36,145.62) -- + (112.58,148.07) -- + (112.79,152.50) -- + (113.00,157.19) -- + (113.21,150.52) -- + (113.43,153.46) -- + (113.64,146.45) -- + (113.85,153.37) -- + (114.06,141.48) -- + (114.28,150.93) -- + (114.49,152.87) -- + (114.70,154.72) -- + (114.91,153.72) -- + (115.13,150.34) -- + (115.34,154.56) -- + (115.55,148.95) -- + (115.76,148.81) -- + (115.98,151.81) -- + (116.19,147.92) -- + (116.40,154.24) -- + (116.61,147.52) -- + (116.82,146.19) -- + (117.04,145.87) -- + (117.25,148.97) -- + (117.46,152.82) -- + (117.67,152.55) -- + (117.89,150.34) -- + (118.10,140.85) -- + (118.31,144.76) -- + (118.52,149.98) -- + (118.74,151.54) -- + (118.95,144.44) -- + (119.16,143.71) -- + (119.37,149.32) -- + (119.59,146.62) -- + (119.80,148.65) -- + (120.01,142.03) -- + (120.22,147.15) -- + (120.44,147.85) -- + (120.65,144.99) -- + (120.86,145.72) -- + (121.07,147.03) -- + (121.29,147.00) -- + (121.50,139.54) -- + (121.71,146.96) -- + (121.92,144.54) -- + (122.14,135.90) -- + (122.35,144.18) -- + (122.56,140.96) -- + (122.77,141.99) -- + (122.99,141.45) -- + (123.20,144.87) -- + (123.41,123.72) -- + (123.62,145.09) -- + (123.84,138.90) -- + (124.05,137.61) -- + (124.26,146.30) -- + (124.47,147.10) -- + (124.69,140.97) -- + (124.90,141.93) -- + (125.11,145.80) -- + (125.32,146.74) -- + (125.54,142.92) -- + (125.75,144.12) -- + (125.96,138.14) -- + (126.17,145.55) -- + (126.39,143.50) -- + (126.60,129.08) -- + (126.81,139.90) -- + (127.02,143.99) -- + (127.23,140.74) -- + (127.45,139.41) -- + (127.66,138.47) -- + (127.87,142.72) -- + (128.08,142.97) -- + (128.30,143.17) -- + (128.51,140.55) -- + (128.72,119.01) -- + (128.93,138.29) -- + (129.15,141.64) -- + (129.36,140.47) -- + (129.57,137.07) -- + (129.78,141.75) -- + (130.00,141.57) -- + (130.21,137.52) -- + (130.42,128.80) -- + (130.63,141.36) -- + (130.85,142.77) -- + (131.06,143.28) -- + (131.27,136.65) -- + (131.48,131.59) -- + (131.70,124.23) -- + (131.91,140.32) -- + (132.12,141.99) -- + (132.33,138.86) -- + (132.55,136.70) -- + (132.76,136.72) -- + (132.97,137.85) -- + (133.18,139.36) -- + (133.40,134.90) -- + (133.61,136.29) -- + (133.82,134.59) -- + (134.03,137.60) -- + (134.25,140.71) -- + (134.46,121.61) -- + (134.67,133.07) -- + (134.88,128.17) -- + (135.10,114.34) -- + (135.31,136.20) -- + (135.52,138.83) -- + (135.73,131.91) -- + (135.95,136.19) -- + (136.16,133.48) -- + (136.37,136.91) -- + (136.58,127.75) -- + (136.79,126.23) -- + (137.01,135.67) -- + (137.22,133.40) -- + (137.43,131.65) -- + (137.64,130.91) -- + (137.86,131.34) -- + (138.07,132.65) -- + (138.28,134.43) -- + (138.49,132.10) -- + (138.71,120.41) -- + (138.92,136.43) -- + (139.13,136.57) -- + (139.34,131.71) -- + (139.56,134.60) -- + (139.77,132.26) -- + (139.98,132.28) -- + (140.19,133.27) -- + (140.41,121.14) -- + (140.62,125.54) -- + (140.83,132.70) -- + (141.04,129.73) -- + (141.26,130.22) -- + (141.47,130.77) -- + (141.68,130.38) -- + (141.89,134.70) -- + (142.11,132.62) -- + (142.32,132.77) -- + (142.53,129.45) -- + (142.74,129.13) -- + (142.96,123.22) -- + (143.17,116.31) -- + (143.38,130.62) -- + (143.59,129.96) -- + (143.81,122.91) -- + (144.02,127.56) -- + (144.23,129.63) -- + (144.44,130.77) -- + (144.66,124.11) -- + (144.87,128.26) -- + (145.08,129.15) -- + (145.29,130.31) -- + (145.51,132.69) -- + (145.72,127.90) -- + (145.93,124.38) -- + (146.14,129.45) -- + (146.36,124.90) -- + (146.57,128.18) -- + (146.78,124.68) -- + (146.99,126.28) -- + (147.20,128.90) -- + (147.42,123.09) -- + (147.63,129.43) -- + (147.84,116.28) -- + (148.05,114.64) -- + (148.27,123.49) -- + (148.48,125.82) -- + (148.69,127.58) -- + (148.90,127.75) -- + (149.12,121.21) -- + (149.33,128.36) -- + (149.54,123.49) -- + (149.75,126.42) -- + (149.97,128.15) -- + (150.18,117.70) -- + (150.39,126.36) -- + (150.60,120.26) -- + (150.82,115.60) -- + (151.03,125.74) -- + (151.24,121.24) -- + (151.45,116.43) -- + (151.67,113.03) -- + (151.88,113.00) -- + (152.09,118.47) -- + (152.30,123.20) -- + (152.52,111.28) -- + (152.73,120.76) -- + (152.94,124.38) -- + (153.15,119.63) -- + (153.37,108.10) -- + (153.58,123.93) -- + (153.79,115.40) -- + (154.00,122.18) -- + (154.22,118.20) -- + (154.43,115.25) -- + (154.64,123.18) -- + (154.85,123.00) -- + (155.07,122.83) -- + (155.28,118.13) -- + (155.49,117.59) -- + (155.70,123.41) -- + (155.92,118.52) -- + (156.13,119.68) -- + (156.34,123.48) -- + (156.55,117.34) -- + (156.77,110.69) -- + (156.98,118.88) -- + (157.19,121.43) -- + (157.40,113.22) -- + (157.61,112.23) -- + (157.83,122.20) -- + (158.04,119.49) -- + (158.25,112.68) -- + (158.46,112.44) -- + (158.68,118.74) -- + (158.89,113.31) -- + (159.10,117.34) -- + (159.31,108.92) -- + (159.53,111.90) -- + (159.74,119.40) -- + (159.95,116.65) -- + (160.16,119.00) -- + (160.38,118.62) -- + (160.59,110.97) -- + (160.80,118.86) -- + (161.01,118.04) -- + (161.23,116.17) -- + (161.44,112.21) -- + (161.65,103.27) -- + (161.86,106.25) -- + (162.08,114.42) -- + (162.29,110.58) -- + (162.50,109.75) -- + (162.71,118.69) -- + (162.93,115.24) -- + (163.14,105.78) -- + (163.35,112.77) -- + (163.56,112.24) -- + (163.78,108.96) -- + (163.99,112.08) -- + (164.20,108.82) -- + (164.41,114.83) -- + (164.63,111.54) -- + (164.84,113.89) -- + (165.05,108.52) -- + (165.26,110.27) -- + (165.48,111.44) -- + (165.69,110.36) -- + (165.90,106.21) -- + (166.11,116.16) -- + (166.33,104.62) -- + (166.54,110.21) -- + (166.75,110.35) -- + (166.96,112.73) -- + (167.18,110.62) -- + (167.39,108.84) -- + (167.60,100.95) -- + (167.81,109.52) -- + (168.02,104.48) -- + (168.24,101.91) -- + (168.45,110.31) -- + (168.66,105.03) -- + (168.87,107.04) -- + (169.09,110.70) -- + (169.30,111.50) -- + (169.51,109.44) -- + (169.72,111.63) -- + (169.94,110.10) -- + (170.15,112.42) -- + (170.36, 92.48) -- + (170.57,102.17) -- + (170.79,106.03) -- + (171.00,107.33) -- + (171.21,109.03) -- + (171.42,109.31) -- + (171.64,107.82) -- + (171.85,106.83) -- + (172.06,100.87) -- + (172.27,100.75) -- + (172.49,104.69) -- + (172.70,106.95) -- + (172.91,101.56) -- + (173.12,106.85) -- + (173.34,100.63) -- + (173.55,106.35) -- + (173.76,107.50) -- + (173.97, 94.36) -- + (174.19,109.39) -- + (174.40,103.76) -- + (174.61,107.95) -- + (174.82,106.14) -- + (175.04,104.43) -- + (175.25,105.40) -- + (175.46,103.68) -- + (175.67, 96.10) -- + (175.89,104.13) -- + (176.10, 99.06) -- + (176.31,104.22) -- + (176.52,103.97) -- + (176.74,101.07) -- + (176.95,106.24) -- + (177.16, 97.91) -- + (177.37, 96.64) -- + (177.59,104.17) -- + (177.80, 96.74) -- + (178.01, 88.67) -- + (178.22, 98.34) -- + (178.43, 88.67) -- + (178.65,104.18) -- + (178.86, 95.60) -- + (179.07,103.84) -- + (179.28,105.25) -- + (179.50,101.71) -- + (179.71, 98.83) -- + (179.92, 96.82) -- + (180.13, 98.70) -- + (180.35, 95.42) -- + (180.56, 97.97) -- + (180.77, 98.55) -- + (180.98,100.23) -- + (181.20, 90.87) -- + (181.41, 98.90) -- + (181.62,103.64) -- + (181.83, 96.40) -- + (182.05,102.64) -- + (182.26,101.89) -- + (182.47, 96.65) -- + (182.68, 99.86) -- + (182.90, 93.47) -- + (183.11, 88.69) -- + (183.32, 95.06) -- + (183.53, 96.46) -- + (183.75, 96.19) -- + (183.96, 91.64) -- + (184.17, 95.51) -- + (184.38, 95.65) -- + (184.60, 99.62) -- + (184.81, 93.91) -- + (185.02, 96.17) -- + (185.23, 99.53) -- + (185.45, 85.77) -- + (185.66, 91.62) -- + (185.87, 97.75) -- + (186.08, 91.07) -- + (186.30, 99.93) -- + (186.51, 95.32) -- + (186.72, 97.82) -- + (186.93, 95.99) -- + (187.15, 87.02) -- + (187.36, 78.91) -- + (187.57, 90.01) -- + (187.78, 93.67) -- + (188.00, 82.85) -- + (188.21, 79.42) -- + (188.42, 70.59) -- + (188.63, 82.86) -- + (188.84, 95.75) -- + (189.06, 90.59) -- + (189.27, 89.02) -- + (189.48, 87.19) -- + (189.69, 91.09) -- + (189.91, 86.22) -- + (190.12, 92.03) -- + (190.33, 94.26) -- + (190.54, 90.43) -- + (190.76, 82.96) -- + (190.97, 87.20) -- + (191.18, 91.88) -- + (191.39, 94.46) -- + (191.61, 88.43) -- + (191.82, 87.20) -- + (192.03, 92.56) -- + (192.24, 91.16) -- + (192.46, 89.69) -- + (192.67, 90.97) -- + (192.88, 88.49) -- + (193.09, 79.23) -- + (193.31, 90.20) -- + (193.52, 92.79) -- + (193.73, 90.04) -- + (193.94, 80.27) -- + (194.16, 87.43) -- + (194.37, 88.65) -- + (194.58, 82.70) -- + (194.79, 80.48) -- + (195.01, 88.23) -- + (195.22, 89.71) -- + (195.43, 91.86) -- + (195.64, 84.25) -- + (195.86, 85.55) -- + (196.07, 88.66) -- + (196.28, 91.60) -- + (196.49, 91.82) -- + (196.71, 84.04) -- + (196.92, 85.74) -- + (197.13, 86.42) -- + (197.34, 81.22) -- + (197.56, 74.16) -- + (197.77, 81.43) -- + (197.98, 73.27) -- + (198.19, 87.06) -- + (198.41, 82.34) -- + (198.62, 83.54) -- + (198.83, 78.85) -- + (199.04, 84.26) -- + (199.25, 75.69) -- + (199.47, 84.88) -- + (199.68, 83.56) -- + (199.89, 75.96) -- + (200.10, 80.43) -- + (200.32, 85.50) -- + (200.53, 82.20) -- + (200.74, 86.09) -- + (200.95, 86.36) -- + (201.17, 86.56) -- + (201.38, 71.72) -- + (201.59, 82.42) -- + (201.80, 80.53) -- + (202.02, 87.48) -- + (202.23, 78.26) -- + (202.44, 78.91) -- + (202.65, 81.75) -- + (202.87, 78.75) -- + (203.08, 81.95) -- + (203.29, 83.10) -- + (203.50, 83.26) -- + (203.72, 84.23) -- + (203.93, 73.79) -- + (204.14, 79.01) -- + (204.35, 84.74) -- + (204.57, 82.12) -- + (204.78, 80.81) -- + (204.99, 78.46) -- + (205.20, 77.64) -- + (205.42, 75.01) -- + (205.63, 72.69) -- + (205.84, 79.61) -- + (206.05, 72.60) -- + (206.27, 71.27) -- + (206.48, 77.10) -- + (206.69, 82.12) -- + (206.90, 72.08) -- + (207.12, 72.22) -- + (207.33, 80.92) -- + (207.54, 77.62) -- + (207.75, 67.69) -- + (207.97, 78.33) -- + (208.18, 73.43) -- + (208.39, 74.19) -- + (208.60, 75.79) -- + (208.82, 83.48) -- + (209.03, 83.46) -- + (209.24, 81.75) -- + (209.45, 79.91) -- + (209.66, 83.75) -- + (209.88, 72.40) -- + (210.09, 81.38) -- + (210.30, 83.70) -- + (210.51, 80.32) -- + (210.73, 84.60) -- + (210.94, 80.90) -- + (211.15, 83.70) -- + (211.36, 85.11) -- + (211.58, 84.45) -- + (211.79, 85.00) -- + (212.00, 85.90) -- + (212.21, 86.41) -- + (212.43, 85.98) -- + (212.64, 85.46) -- + (212.85, 86.13) -- + (213.06, 86.76) -- + (213.28, 87.48) -- + (213.49, 86.86) -- + (213.70, 87.73) -- + (213.91, 87.85) -- + (214.13, 88.88) -- + (214.34, 88.79) -- + (214.55, 89.06) -- + (214.76, 89.58) -- + (214.98, 89.76) -- + (215.19, 89.97) -- + (215.40, 90.59) -- + (215.61, 90.69) -- + (215.83, 91.24) -- + (216.04, 91.37) -- + (216.25, 91.73) -- + (216.46, 92.06) -- + (216.68, 92.45) -- + (216.89, 92.85) -- + (217.10, 93.17) -- + (217.31, 93.48) -- + (217.53, 93.78) -- + (217.74, 94.17) -- + (217.95, 94.52) -- + (218.16, 94.83) -- + (218.38, 95.13) -- + (218.59, 95.50) -- + (218.80, 95.82) -- + (219.01, 96.19) -- + (219.23, 96.50) -- + (219.44, 96.83) -- + (219.65, 97.17) -- + (219.86, 97.49) -- + (220.07, 97.82) -- + (220.29, 98.18) -- + (220.50, 98.48) -- + (220.71, 98.84) -- + (220.92, 99.17) -- + (221.14, 99.51) -- + (221.35, 99.83) -- + (221.56,100.17) -- + (221.77,100.51) -- + (221.99,100.85) -- + (222.20,101.17) -- + (222.41,101.51) -- + (222.62,101.86) -- + (222.84,102.19) -- + (223.05,102.52) -- + (223.26,102.86) -- + (223.47,103.20) -- + (223.69,103.53) -- + (223.90,103.87) -- + (224.11,104.20) -- + (224.32,104.54) -- + (224.54,104.87) -- + (224.75,105.21) -- + (224.96,105.54) -- + (225.17,105.88) -- + (225.39,106.21) -- + (225.60,106.55) -- + (225.81,106.89) -- + (226.02,107.22) -- + (226.24,107.56) -- + (226.45,107.89) -- + (226.66,108.23) -- + (226.87,108.56) -- + (227.09,108.90) -- + (227.30,109.23) -- + (227.51,109.57) -- + (227.72,109.90) -- + (227.94,110.24) -- + (228.15,110.57) -- + (228.36,110.91) -- + (228.57,111.24) -- + (228.79,111.58) -- + (229.00,111.92) -- + (229.21,112.25) -- + (229.42,112.59) -- + (229.64,112.92) -- + (229.85,113.26) -- + (230.06,113.59) -- + (230.27,113.93) -- + (230.48,114.26) -- + (230.70,114.60) -- + (230.91,114.93) -- + (231.12,115.27) -- + (231.33,115.61) -- + (231.55,115.94) -- + (231.76,116.28) -- + (231.97,116.61) -- + (232.18,116.95) -- + (232.40,117.28) -- + (232.61,117.62) -- + (232.82,117.95) -- + (233.03,118.29) -- + (233.25,118.62) -- + (233.46,118.96) -- + (233.67,119.29) -- + (233.88,119.63) -- + (234.10,119.97) -- + (234.31,120.30) -- + (234.52,120.64) -- + (234.73,120.97) -- + (234.95,121.31) -- + (235.16,121.64) -- + (235.37,121.98) -- + (235.58,122.31) -- + (235.80,122.65) -- + (236.01,122.98) -- + (236.22,123.32) -- + (236.43,123.65) -- + (236.65,123.99) -- + (236.86,124.33) -- + (237.07,124.66) -- + (237.28,125.00) -- + (237.50,125.33) -- + (237.71,125.67) -- + (237.92,126.00) -- + (238.13,126.34) -- + (238.35,126.67) -- + (238.56,127.01) -- + (238.77,127.34) -- + (238.98,127.68) -- + (239.20,128.01) -- + (239.41,128.35) -- + (239.62,128.69) -- + (239.83,129.02) -- + (240.04,129.36) -- + (240.26,129.69) -- + (240.47,130.03) -- + (240.68,130.36) -- + (240.89,130.70) -- + (241.11,131.03) -- + (241.32,131.37) -- + (241.53,131.70) -- + (241.74,132.04) -- + (241.96,132.38) -- + (242.17,132.71) -- + (242.38,133.05) -- + (242.59,133.38) -- + (242.81,133.72) -- + (243.02,134.05) -- + (243.23,134.39) -- + (243.44,134.72) -- + (243.66,135.06) -- + (243.87,135.39) -- + (244.08,135.73) -- + (244.29,136.06) -- + (244.51,136.40) -- + (244.72,136.74) -- + (244.93,137.07) -- + (245.14,137.41) -- + (245.36,137.74) -- + (245.57,138.08) -- + (245.78,138.41) -- + (245.99,138.75) -- + (246.21,139.08) -- + (246.42,139.42) -- + (246.63,139.75) -- + (246.84,140.09) -- + (247.06,140.42) -- + (247.27,140.76) -- + (247.48,141.10) -- + (247.69,141.43) -- + (247.91,141.77) -- + (248.12,142.10) -- + (248.33,142.44) -- + (248.54,142.77) -- + (248.76,143.11) -- + (248.97,143.44) -- + (249.18,143.78) -- + (249.39,144.11) -- + (249.61,144.45) -- + (249.82,144.78) -- + (250.03,145.12) -- + (250.24,145.46) -- + (250.45,145.79) -- + (250.67,146.13) -- + (250.88,146.46) -- + (251.09,146.80) -- + (251.30,147.13) -- + (251.52,147.47) -- + (251.73,147.80) -- + (251.94,148.14) -- + (252.15,148.47) -- + (252.37,148.81) -- + (252.58,149.15) -- + (252.79,149.48) -- + (253.00,149.82) -- + (253.22,150.15) -- + (253.43,150.49) -- + (253.64,150.82) -- + (253.85,151.16) -- + (254.07,151.49) -- + (254.28,151.83) -- + (254.49,152.16) -- + (254.70,152.50) -- + (254.92,152.83) -- + (255.13,153.17) -- + (255.34,153.51) -- + (255.55,153.84) -- + (255.77,154.18) -- + (255.98,154.51) -- + (256.19,154.85) -- + (256.40,155.18) -- + (256.62,155.52) -- + (256.83,155.85) -- + (257.04,156.19) -- + (257.25,156.52) -- + (257.47,156.86) -- + (257.68,157.19) -- + (257.89,157.53) -- + (258.10,157.87) -- + (258.32,158.20) -- + (258.53,158.54) -- + (258.74,158.87) -- + (258.95,159.21) -- + (259.17,159.54) -- + (259.38,159.88) -- + (259.59,160.21) -- + (259.80,160.55) -- + (260.02,160.88) -- + (260.23,161.22) -- + (260.44,161.55) -- + (260.65,161.89) -- + (260.86,162.23) -- + (261.08,162.56) -- + (261.29,162.90) -- + (261.50,163.23) -- + (261.71,163.57) -- + (261.93,163.90) -- + (262.14,164.24) -- + (262.35,164.57) -- + (262.56,164.91) -- + (262.78,165.24) -- + (262.99,165.58) -- + (263.20,165.91) -- + (263.41,166.25) -- + (263.63,166.58) -- + (263.84,166.92) -- + (264.05,167.26) -- + (264.26,167.59) -- + (264.48,167.93) -- + (264.69,168.26) -- + (264.90,168.60) -- + (265.11,168.93) -- + (265.33,169.27) -- + (265.54,169.60) -- + (265.75,169.94) -- + (265.96,170.27) -- + (266.18,170.61) -- + (266.39,170.94) -- + (266.60,171.28) -- + (266.81,171.61) -- + (267.03,171.95) -- + (267.24,172.28) -- + (267.45,172.62) -- + (267.66,172.95) -- + (267.88,173.29) -- + (268.09,173.62) -- + (268.30,173.96) -- + (268.51,174.29) -- + (268.73,174.63); +\end{scope} +\begin{scope} +\path[clip] ( 0.00, 0.00) rectangle (325.21,216.81); +\definecolor{drawColor}{RGB}{0,0,0} + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 56.49, 48.00) -- (254.58, 48.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 56.49, 48.00) -- ( 56.49, 42.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 84.79, 48.00) -- ( 84.79, 42.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (113.09, 48.00) -- (113.09, 42.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (141.38, 48.00) -- (141.38, 42.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (169.68, 48.00) -- (169.68, 42.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (197.98, 48.00) -- (197.98, 42.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (226.28, 48.00) -- (226.28, 42.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (254.58, 48.00) -- (254.58, 42.00); + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 46.83, 26.40) {10}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at ( 56.82, 30.49) {-17}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 75.12, 26.40) {10}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at ( 85.12, 30.49) {-15}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at (103.42, 26.40) {10}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at (113.42, 30.49) {-13}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at (131.72, 26.40) {10}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at (141.72, 30.49) {-11}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at (161.77, 26.40) {10}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at (171.76, 30.49) {-9}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at (190.07, 26.40) {10}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at (200.06, 30.49) {-7}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at (218.36, 26.40) {10}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at (228.36, 30.49) {-5}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at (246.66, 26.40) {10}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at (256.66, 30.49) {-3}; + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 56.49, 48.00) -- (254.58, 48.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 56.49, 48.00) -- ( 56.49, 42.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 84.79, 48.00) -- ( 84.79, 42.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (113.09, 48.00) -- (113.09, 42.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (141.38, 48.00) -- (141.38, 42.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (169.68, 48.00) -- (169.68, 42.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (197.98, 48.00) -- (197.98, 42.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (226.28, 48.00) -- (226.28, 42.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (254.58, 48.00) -- (254.58, 42.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00, 53.81) -- ( 48.00,187.83); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00, 53.81) -- ( 42.00, 53.81); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00, 76.15) -- ( 42.00, 76.15); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00, 98.48) -- ( 42.00, 98.48); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00,120.82) -- ( 42.00,120.82); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00,143.16) -- ( 42.00,143.16); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00,165.50) -- ( 42.00,165.50); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00,187.83) -- ( 42.00,187.83); + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 33.60, 44.14) {10}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at ( 29.51, 54.14) {-12}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 33.60, 66.48) {10}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at ( 29.51, 76.48) {-10}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 33.60, 90.57) {10}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at ( 29.51,100.57) {-8}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 33.60,112.91) {10}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at ( 29.51,122.90) {-6}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 33.60,135.24) {10}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at ( 29.51,145.24) {-4}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 33.60,157.58) {10}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at ( 29.51,167.58) {-2}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 33.60,181.09) {10}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at ( 29.51,191.08) {0}; + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00, 53.81) -- ( 48.00,187.83); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00, 53.81) -- ( 42.00, 53.81); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00, 76.15) -- ( 42.00, 76.15); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00, 98.48) -- ( 42.00, 98.48); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00,120.82) -- ( 42.00,120.82); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00,143.16) -- ( 42.00,143.16); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00,165.50) -- ( 42.00,165.50); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 48.00,187.83) -- ( 42.00,187.83); +\end{scope} +\begin{scope} +\path[clip] ( 48.00, 48.00) rectangle (277.21,204.81); +\definecolor{drawColor}{RGB}{0,0,0} + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 84.79,123.38) {Round-off error}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 84.79,111.38) {dominant}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at (240.43,123.38) {Truncation error}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at (240.43,111.38) {dominant}; +\definecolor{fillColor}{RGB}{255,255,255} + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 56.49, 87.31) rectangle (177.75, 51.31); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 65.49, 75.31) -- ( 83.49, 75.31); + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 4pt off 4pt ,line join=round,line cap=round] ( 65.49, 63.31) -- ( 83.49, 63.31); + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 92.49, 71.87) {Forward difference}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 92.49, 59.87) {Center difference}; +\end{scope} +\end{tikzpicture} diff --git a/doc/Articles/Autodiff/figures/backprop/backprop-eps-converted-to.pdf b/doc/Articles/Autodiff/figures/backprop/backprop-eps-converted-to.pdf new file mode 100644 index 000000000..a76f7346c Binary files /dev/null and b/doc/Articles/Autodiff/figures/backprop/backprop-eps-converted-to.pdf differ diff --git a/doc/Articles/Autodiff/figures/backprop/backprop.eps b/doc/Articles/Autodiff/figures/backprop/backprop.eps new file mode 100644 index 000000000..a332cb3ef --- /dev/null +++ b/doc/Articles/Autodiff/figures/backprop/backprop.eps @@ -0,0 +1,650 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: cairo 1.8.8 (http://cairographics.org) +%%CreationDate: Tue Jan 23 23:31:51 2018 +%%Pages: 1 +%%BoundingBox: 1 1 377 199 +%%DocumentData: Clean7Bit +%%LanguageLevel: 2 +%%EndComments +%%BeginProlog +/cairo_eps_state save def +/dict_count countdictstack def +/op_count count 1 sub def +userdict begin +/q { gsave } bind def +/Q { grestore } bind def +/cm { 6 array astore concat } bind def +/w { setlinewidth } bind def +/J { setlinecap } bind def +/j { setlinejoin } bind def +/M { setmiterlimit } bind def +/d { setdash } bind def +/m { moveto } bind def +/l { lineto } bind def +/c { curveto } bind def +/h { closepath } bind def +/re { exch dup neg 3 1 roll 5 3 roll moveto 0 rlineto + 0 exch rlineto 0 rlineto closepath } bind def +/S { stroke } bind def +/f { fill } bind def +/f* { eofill } bind def +/B { fill stroke } bind def +/B* { eofill stroke } bind def +/n { newpath } bind def +/W { clip } bind def +/W* { eoclip } bind def +/BT { } bind def +/ET { } bind def +/pdfmark where { pop globaldict /?pdfmark /exec load put } + { globaldict begin /?pdfmark /pop load def /pdfmark + /cleartomark load def end } ifelse +/BDC { mark 3 1 roll /BDC pdfmark } bind def +/EMC { mark /EMC pdfmark } bind def +/cairo_store_point { /cairo_point_y exch def /cairo_point_x exch def } def +/Tj { show currentpoint cairo_store_point } bind def +/TJ { + { + dup + type /stringtype eq + { show } { -0.001 mul 0 cairo_font_matrix dtransform rmoveto } ifelse + } forall + currentpoint cairo_store_point +} bind def +/cairo_selectfont { cairo_font_matrix aload pop pop pop 0 0 6 array astore + cairo_font exch selectfont cairo_point_x cairo_point_y moveto } bind def +/Tf { pop /cairo_font exch def /cairo_font_matrix where + { pop cairo_selectfont } if } bind def +/Td { matrix translate cairo_font_matrix matrix concatmatrix dup + /cairo_font_matrix exch def dup 4 get exch 5 get cairo_store_point + /cairo_font where { pop cairo_selectfont } if } bind def +/Tm { 2 copy 8 2 roll 6 array astore /cairo_font_matrix exch def + cairo_store_point /cairo_font where { pop cairo_selectfont } if } bind def +/g { setgray } bind def +/rg { setrgbcolor } bind def +/d1 { setcachedevice } bind def +%%EndProlog +%!FontType1-1.1 f-0-0 1.0 +11 dict begin +/FontName /f-0-0 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/FontBBox {0 -250 703 750 } readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 1 /g1 put +dup 2 /uni0061 put +dup 3 /uni0029 put +dup 4 /uni0046 put +dup 5 /uni006F put +dup 6 /uni0072 put +dup 7 /uni0077 put +dup 8 /uni0064 put +dup 9 /uni0070 put +dup 10 /uni0073 put +dup 11 /uni0062 put +dup 12 /uni0042 put +dup 13 /uni0063 put +dup 14 /uni006B put +readonly def +currentdict end +currentfile eexec +f983ef0097ece636fb4a96c74d26ab84185f6dfa4a16a7a1c27bbe3f1156aea698df336d20b467 +b10e7f33846656653c5ac6962759d3056cbdb3190bac614b984bf5a132dc418192443014ba63de +800a9c2cbaae3e98d910ec768e3ec0a68d6662ed0a4786decd0bc494681b7a8f496910cd7b4630 +6dc8ae9298e870deef7a0ea25578226e409aa0b4147f7b82b7598ad503870b1045c168eb0c100e +42b0f810d306f8f7f38011c973c0fb43b3d4511ebd07847dd8587bb33de670b6b0b94ed7772fe2 +92f5b1bfe83ad2c9d4f17ed4f34a3ab5b1c268ae43a5947d16525f9ffef6d057d70f07acc67574 +27d7d3f6aa307d2bc8f59c82b8579918945de0cbfe985b93304f9c14bf82fa81a93da770200793 +2dd1508907c081ece0f1ee6c8b723038fa488f2c81ddbb1ed051535205378c6d03a283ffea585c +6512c1ac0452fb99c4190b42973a1583935f2b9ecfda4f5c6879c310a39088a1c9f6c7d48e44ff +eff56f83cb68d530cc40ea36342fdb74d52256cb883f6759820be5b0f9a34c5da927f964272e52 +dc34b64ab876eaebf2e744f2f0d8f5845003e27a3a27e098f035c208639194ae202e355a00fea8 +136991d4991f2536e7f6bde1bc7143538d29199f4f20230fd5ecb614ab715150d159a0918fcfb6 +358f638698f9442655d673f28397c64f0cad2df281ad1308353e3583992169fe2170384c6d33ce +abb707113d40c13481d0b05be5614b687e83d58bfdd52339f971d1edcb77cf95f6520452ec32bf +d5261a0c0e07d7992c376a677fcf40e4fd044ba412e003d0332d9b1f1b0e4456b089ce9837d0f1 +15bd55065e60319b0a843800ca3b9d6fdc794179b340d1e77ce705a439ca07daccab58f9186e94 +cdaa29beb312cd97993f8a353c705a405f445106127a7f902199a9dd5dcbb2329be7f2f40125e0 +3d22c73b271e28479d0194cd949e8848c243ba17c2d37fbb86326e12840dfcaaaf5bdcda5bb181 +1800aa2938e98ee8b67d291bd5e5b95d8e7dd36b4a71da9486eacc2e89923792520818cebdb349 +ca34479be318df799d2f42fd59836e3422caa3c25a647622e0da55d98ba7591408713aba7b40c5 +b8e3cbb63a5552ca72bf360dcd2ae223ef68c37ce2f8bc494e05805e5bacc706190fcf693c80eb +63f77aa65289faa69895f49e491410c6d5f1cc2300a9012dbfd915937f2bc8bdc4ad7a5bdc0a9a +a6dcec317f94a4d22bc1aa1464fa30132bb897136957f69f80e4b9fe051b9552da17dd4d776328 +bbe9db2c435d733d0023160559e903dd283d7d834c3a5117b67919e9888d80b4596c1a95ff11b5 +8fd6f817680add6d340040e497d0b5a326d4eb4221b8359a72da0bd0849d4aea1c5d66d9ef5c80 +a068411472672882862c9cc6e67ec4f3d80d0b4e10645da2edfa586183e1f5884b3413339f7d36 +4dfe9771fa7641efe09542369e3d2a67631c16c9db4ad96f58a937e84d390fc3929edb40453fa4 +9ce08f6f4f9cc828b510547000d0b1950a9df6893395c4224eb4bb09c9cc952f1951cc7697df04 +5f33f0445361bda924fab0c96ef285a87766237f3153020ab09bcfe9d6ab2b17ce3eb94a28598d +c905592fb8ff800170a36cb22e54dba7bf25168084e9914ed68f9d8564876c5eb8c4d1cf696141 +720948c7fff51dd431baa98edc00aada78d4a94cef0a01434f3f57724b5cfd426c79feb930a95d +7c886a6e427726d0cfedd5060d6b432c92c1bd752f6ac6f2df5bd07b53cbed80f1413e41e06284 +3288de3c08a038d0741b410f60aa4071767129604ff540d876e258afec2d64f00f179e147b0fb9 +43101e2391840e8b507a8c4bacd82f2ac158a72eb06aeea7523eaf286baaf0e9558f13886c69b0 +de61bb64126986e1a3c3b20337a806144051f9c0aa1a42a05b2c599ca827d687f4133e1e7d7776 +8d1014b8c07ab69b56bfb9e7edb5d7ab66fd739077f910fec38e75af74302053bbf174be79f719 +9bf0185ce294137a2804878c20c647246c37b367a0a96171e0a417866cab70bf94da9e0eac092e +169d44bd37d22b47a69de9ed3c0948051a8aa4de3ebf65b2cfa78361502261e98be16281c2541e +58d2a2624603b25bb01a237207092c0b2fe1bcd98f921b4ab15e72b2c08385d9b7375de6c82f7c +0ce46d643c85f229100162735f51ecf749bb51039fadf1b8d166a0cff86aae824ba1fec3845eac +1b6f50c3ec9bd0719a18651372a05e618e5a7d11bd8f359b0705ecf396d8856b3cced22f428ba6 +78b3dea13299710ff7028785510f5519acf2cfa2b9eacd17963afd391c011d3710848500b284b3 +ca2f0dcc9cb7e54434d088900be49c396f0b1db2a625fcd16fe4f9c28147ededf18508ded01053 +e8830f6a6c9102bb5aa91ec9a47cf59d00e4abc987b5560f43f5ebc098757b3eb010d0775dd761 +802bbabe16934ced432a9113dfbc89cde064557eecaaad8b65fd7390e455699cf0a66211d5b232 +45f13e968775359e777981ccf4c08611f266e9b45e2e897d229e82d61b75481d626c63696108c7 +fef9d5a03c9bcbf4f19e68f8185ab67090ea782303f03f5cdc3db65636d347c4ebe13a5e553e54 +b0d11e53f9765a0caa21417471d0bb861beb2e57fb62c051d2e241ff524b6de48cffa72dca1868 +1173d8b1626e68e9438be808ea73060264a0588320380c53fd4e22aced9b58b73a02c7c7844e39 +69bb647a58a484aba8f05658aba1f1ac4f6cd1defd2efb330982db6d2ad76baa3cc500589c2346 +6a7e5930e2cf7a312348797ad03fb006d5a956d8aea7648b791c0d1b6e7378822a7d31d9c8985c +72b79d0f4e8da8c4b1085785d1d74962f96406156ce2f8b2ce1e1e9eb0f2842d69a8e9ff0427c3 +502adc1c5884f85dbfa44d2f9c0d4e5dd6f3d9997983c8cee03d8c02ce9f0aad219b159113b6ab +01368ac92822dc00af917a4a0514e70228ecc2293f57ba9a3daca3db6997f4d55d7e0dcd1cf52a +c9323b2529b10eaf558bd1271dcf42692778d144bc2df03a52e7bc25d4717dfa93f75bb4a04c3d +fd8a4c35253b6acb0217f7a88036fbb1f2e74b4a6063aee99612adf3066d2ba977eda6c39ff806 +a07d639f985d82bd5850ef3b12d8224b95b440179ddab8844290677b425a17e071e0d0cba5de07 +1bb6cc188343615cb79370604e57fb04eae8e8d5f325410f809c4bd63df311771fbad98f606e89 +a2028e3ab1be4cc0f97c8817cd5b9a7de239b3d9603d34ff03a4a3c86302ab43ad7f583e83e51a +45e6ed849060cd6a442496141d247dde24be1c8c8440c15ebe12de69ef8eb626181537b1286274 +b856549bb065d6f092a9823da544cf378726703359e0feda8063cddf141d68bb486c0b5085527c +2deb51c15f2da1d760182ecd326461ea7f206c39debdad7d0cebc97f8f40da2236776b1087258d +d07e3176b0bea9b6b316c510bae30fbedd118b36cbd5d813c5cdb6b67fe6ef28b5488f8dc51309 +91216ff807199e36c98b1d1183a36a645b69a89450bf33792cf825b41e8f63bd4a978445b90ba5 +5c71b5c631904d2f10ec6090e33fb3fdfee2d940c5940e81f50c9e4923a3af72cdebeab8511f98 +7e96279e964df28f4ac9069a423f7f44dbc3b0ac3232298429622bffce0bfccc81a9c1b6934ba5 +d094f10255651cfac77028a67e1a6df71c74066468f2b6e318a761ad0ec8e727daf45361c0a22e +1bf1cde946209cc355535b7ad556edb98998800db13ccc7058846cab6062a9ddb5f74724c729cb +5e4ac8f6134bccbfe336125b8e4a78e5bfb7766e0bc13a4bc604688c82323342b118c13bdf08e0 +d8ec40f1a4e902157233e95823872f08a114d0bc634e7b9653d2d918c912c549ae0b57f7b40f7c +1fc41b4496151a257b761323ee9b206709adbc58911e5f53dc14eac7d0c4423d10f8b9c1bb275d +239885a0ca26cbee44a461749fc2e92fa953e9c6a4bce9a5768fd636598ecd5985d2adc4df4ba6 +52d393a7e007bb1a64aa8ca95ca355bc170000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +cleartomark +%!FontType1-1.1 f-1-0 1.0 +11 dict begin +/FontName /f-1-0 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/FontBBox {0 -250 766 750 } readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 1 /uni0078 put +dup 2 /uni0045 put +dup 3 /uni0079 put +dup 4 /g4 put +dup 5 /uni0074 put +dup 6 /uni0040 put +dup 7 /uni003D put +dup 8 /uni0077 put +readonly def +currentdict end +currentfile eexec +f983ef0097ece636fb4a96c74d26ab84185f6dfa4a16a7a1c27bbe3f1156aea698df336d20b467 +b10e7f33846656653c5ac6962759d3056cbdb3190bac614b984bf5a132dc418192443014ba63de +800a9c2cbaae3e98d910ec768e3ec0a68d6662ed0a4786decd0bc494681b7a8f496910cd7b4630 +6dc8ae9298e870deef7a0ea25578226e409aa0b4147f7b82b7598ad503870b1045c168eb0c100e +42b0f810d306f8f7f38011c973c0fb43b3d4511ebd07847dd8587bb33de670b6b0b94ed7772fe2 +92f5b1bfe83ad2c9d4f17ed4f34a3ab5b1c268ae43a5947d16525f9ffef6d057d70f07acc67574 +27d7d3f6aa307d2bc8f59c82b8579918945de0cbfe985b93304f9c14b9ebce3f978066a15e8e1c +b8872669361a9912c11ae5a12685e20e3a574fb8f5c503c55bdc4f2f97b63599b38a4b82078158 +6824607f86ed6a4e2550c5a80c5c7238ab889d73953a4fbeae2b569260d980e20d6ac48f9813ba +d76f7624f1a9776f46fa475931f4acdb576a47b2db103722545b54f191651f855c67233c270449 +688ae651b88d78c8972bf0ff8b6496cad1e36d3e70284b45de7161ae3788d098898aa40327d971 +35861e5217018a802384bc6f712d6325f3a1e8c6c1ab7b79abce11cf82d88cd1b5eda2d146f6a5 +cc6c2a29dd858b906d9ee39d6a18cf70b4b2b793b9371d594dc81a27ab5023bbf158e07b1ea085 +ce28f7b81902277c241ecb16a8930f5d06cedfd905334cde1524e15f619ce117bd3df9f9766404 +2370ffc21dd784b576afcb1e362564481c30c72deb8fcda4f0323901eaf210470211fb872e1115 +5b233a2a330e13a961528d201baccde13309d62ea565cc874fbd88934c9cbec1499f8b19d65d0e +843ec372fb5369864328f57be3e321d286d1264f0343d1916ba2384d6ee269ae7b6eae2b372103 +ca9742c0091d9f0ad3332a19839d2be908816cea939182b79e8268b18b6798af21976d47ded678 +7bf13e3b539bc51b7d92792299719de3b8fe3c0816df7db52c04c022561c1d97e83048f450e9ab +db2835f9906edea4f4c47edc57a4fb2f257c17a07754b37a02fb73ba91dad8d4ba8f48c0bb1324 +ca13b442cd9614b922415e37267f83466f4f832b56cea8c0b144881b08aca238f23e6d2f892885 +3d1786304224452000b8c5bd8b5e9d49d98411a0ca6e1bdb6d362eeba41cb0fa6aca537c2d244d +a048a719446d9562677f47386a45e5ebc9421ad85de3e0473e7a29ea1a423dd4ab10fabb2f95ea +6e371f0e7ed0a66335284f4a25a6842f835662d928c37a5913607ca98ae1c48be483cc31969d36 +34a2545101541ab389d52ef8d3324016ffb72f99f4705022eeed8c9560086937d84d57af894f03 +7dff92cf9f83c3b67374a05171a141423e8bf5d552a808d0c671fe16bdb64f17d26dce9d9ddb91 +cb04863eaaf1390aa5602e1a91532caff41b3d45639964bb765f3d6ac92bb8b9105bb18cb73301 +65b4ffa03bcc1217576991d3c7bebfbef10627df4e685309444cd3fa92ca9fe80754863b06cf2a +36df89af19fc3f581f03bbb8c535f978416fd2fc56fcdba67791c5343420d9e85530127f0a5c8b +05c89ed37d97dd69adbf30262ac8b9db752181d9c51a4dfc04b63bbf98c7c4b413fdfbc101550f +af889cc30c445ebbe86737e7257bf4657cbcd0d6828536fc51836ff3eab24c7878867bfe9ad577 +c28d4d064d5e1a51934e592a21739a53c137af4c79afd1940a94d004b8f9af2f56921500c0c5e4 +37ada19550294f554c520e7d232311854c579c3db90ddd720cb7bfda6495d80040be25ce92d292 +b0b72f8267da6e7861b152454c46b9bdf334a5d20d15e270bed5f5920d6085aa85c10b3704255e +fb9d4637c68fe595b0db73e03cd8dff167d9205a6e26275da84b7c5089fe3984be304ee8645df9 +519be21afe772252524b32d2cf4585258ffc91f7eb6579478789219263f40c72d798fc1e1a9b8f +c8cf3f05fabaec84a2ae2c8cb596344996abc2707457618cc6d9b5b98090f2d52aa72f5c7493fe +5d0bc62d690cf0cabd64d2c4bb223c2575c93288adf6106f6ca36f574d037a7fa6f0d2dd784191 +302f2b16b138f7d67048b3a4eceb007af35c009873a9574e88256e33b85cef4b2cc4d89e0fd74a +6f7a407267bf8a04b3e6867b38d0db2c96ff9ffe6262a2ada612e101a6c314d0955d5f13ba8712 +48226fb301483ac93da9cc80e4da8d7d6fe69a41eeb19e91915599799027b7fff39121e591b700 +db385f4afdf7e1e6325529552decc00ce96b0c8a7fbe97c55f3c3050b289a76e6233ff4f04f7f3 +3894220591150f0bcefc76c0cc46043fc8ed2ae52e1b9ab09f928246f11d63645b01e59705d97c +aa32bf9d5c2979779aeea89b2d45411ceb451a244ce83154a0903729972246982e8830f5f870f6 +648eca98f462471f4e964a2cf3bb286bb21dad73d1cd73ecd5435b21da80f112be0c3e70835357 +16193f81afa2a6c3f76b5e4932da2dbc851ffcbd44617b7e1d1b38e6989c3dab8f8d98f3b038bc +69a660d2b30d37e636c4b9d5a9e5488fa3c2d52c20d6518e0a78683e0d1001da23e0341001a1e6 +a6a704f7f561009a55fda5191b240819503312a100e2597c6d90a0c08ae4168031e4b9e705fd50 +0477a439b58c3e814f4faf005612c7ca9b7f9ea629367177da069eab27a75d9390b8296ce3f1d5 +6103daf4207001f915c70b25c404df6770f49cd1528fbe4eace30f55e4d9a8a8310dc0afc942a0 +78d128d117adeb69d8530c9d8b6109a69469617c2ba0c3289e63f300a41d0c74ba5991491e6286 +268e49e5815a7199cd7762303bbd2366c1821e64ce1a1ac6655f50d61cdcc87240ef15d9de6bd7 +c8530000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +cleartomark +%!FontType1-1.1 f-2-0 1.0 +11 dict begin +/FontName /f-2-0 def +/PaintType 0 def +/FontType 1 def +/FontMatrix [0.001 0 0 0.001 0 0] readonly def +/FontBBox {0 -20 529 674 } readonly def +/Encoding 256 array +0 1 255 {1 index exch /.notdef put} for +dup 1 /g1 put +dup 2 /uni0032 put +dup 3 /uni0033 put +dup 4 /uni0034 put +dup 5 /uni0035 put +dup 6 /uni0036 put +readonly def +currentdict end +currentfile eexec +f983ef0097ece636fb4a96c74d26ab84185f6dfa4a16a7a1c27bbe3f1156aea698df336d20b467 +b10e7f33846656653c5ac6962759d3056cbdb3190bac614b984bf5a132dc418192443014ba63de +800a9c2cbaae3e98d910ec768e3ec0a68d6662ed0a4786decd0bc494681b7a8f496910cd7b4630 +6dc8ae9298e870deef7a0ea25578226e409aa0b4147f7b82b7598ad503870b1045c168eb0c100e +42b0f810d306f8f7f38011c973c0fb43b3d4511ebd07847dd8587bb33de670b6b0b94ed7772fe2 +92f5b1bfe83ad2c9d4f17ed4f34a3ab5b1c268ae43a5947d16525f9ffef6d057d70f07acc67574 +27d7d3f6aa307d2bc8f59c82b8579918945de0cbfe985b93304f9c1debee09fd9cb05390ad5555 +1edc479557e28c77bf40ae0ca1e76c9e9d5e9fd505cb2d360d436ba51b1eb6a57ed14c172717fa +9bb2c7a10c31769dc26929a055f956f7cf149ab4cba49646d92263d9bfee7cea7d985c42241e6e +c88d5f4d218a47241552eb4d05d25b1cb159184c752d34efb284de3305f87a7be8099e4f7c56f7 +ae7e6b54f225ee0622bf95bb82bcc6d918117844dcf54aded8483f8e00fd0468defec73a6e95fc +3db701aafe06cbc8ea4d7bd4bfb424d91dabdca38f7ad5a466cec9679a751e60f09882d0e8bbcb +b2d556f429c1120168384e48feea2883b6c79384234f3048bac469d1f48d65101201e176a818a1 +c1730e3843bae40fe84eb6c321482354833dcb3fe914ee9284b94942ee8bce81b5ca69224011d1 +a4afd56e3806faa7fda0078a5c112add219abba1a16346ce5cc904dfaca53701e132fef0eb59a8 +de7335876f57d22ae2b3ea046513df283556be1a4bc8d08ff28a80d58158e93125060fe79f22fe +dd8d269a5720c0ec14b139b0ff895b79bed2d383e5b7c5ec615b697020e344c60310c80a035cfe +2ffaf97d111e289f4bbd47f2f483dd1fec2a50ba7d809c810e91c3428d0e8194e7f531704eb45a +8fb198386426fa42a9baf9220254609045953f2fdc846b521d8e4c4c66f9dadd67267499b2833c +41d5037f6455b6d99c5be4bfaa07f03be8b476a0ad2bdf1b5a7224f452398a61aa7faaae6e9b71 +12257c51bf7274c7a3859f6db6228c0105aee63991e71a7573b8bfe4db74ce1f993fb138481852 +176e65a7b900fdd664707d931ad887be0401b89a34f3f049dd1c01877bc862a30b9b5fc78f8514 +88c9eadf68ff7195898a9fea872ef6da1c6869db919d3319c875eab952f04dd48115643d9ee173 +f76d6571ec9a297ce679dd1465de3d796dfe35a0ef8edb13fae907d4db3af55bd738245db5047f +6cbfdab3ed718cc8af3c907396c0c6b7b3396e70363660322617460417b9ee73ec79b1fe34ab13 +48e13b679d43f9ba5e299d877983c8c75e4445628d9af01adbf0cd6b434c83657b97681dd6971a +aaab2d91744913c484a43b4b01aea8b901892752d5df833c367a9e7a475cd2bbd9902d82be506b +2adc3df42ad4538e986630a1964d1aac90cb949f0b50bbf516a8a4500eb0b4b516b826190e2fca +a96713f071d113b4769a25100e9ccbe77fb69d700e957ef5bf8037ce5a5bb6fdf6c5e699921ab3 +cf705b02f235944d634af833f1a7f7c2b02657fa5ed68ced32b34bd8d1a65ac2ae71aef98489ee +79a0f205d2f8a683f1bae9cc4d6fe79ac36062dff0e27bb082914e1a762bc6377b5797e7ea2bf2 +8a0262632e9a49238a8ce58c222bee6836729041561ff67c310631634eba45e2c5208bef811ba1 +cd32e0725a672e9ddc3f9330ce19fef56b7ff88b2262e646c760a017aa87039c5bd16b00c36081 +626c5adb38b08d83f6496d968384eaea9e153ce52120b3b2cb7a3d37b6003807e71638be92c3c8 +8809a0fd54f53b0202d17232493d783b62b8824ab9df7a4aa536d37669c969a09ede073c66fcd7 +498a3ec623ed603c0094862f5a589e7890a875a273394decdd423f0214a8ec6500e04b6a89b8dd +e284f787adec90657bcfedfbd551a348bc200af24ecb2a7f66cb0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +0000000000000000000000000000000000000000000000000000000000000000 +cleartomark +%%Page: 1 1 +%%BeginPageSetup +%%PageBoundingBox: 1 1 377 199 +%%EndPageSetup +q +0 g +0.4 w +0 J +1 j +[] 0.0 d +10 M 112.508 136.895 m 115.473 142.82 113.07 150.031 107.141 152.992 c +101.215 155.957 94.008 153.555 91.043 147.629 c 88.078 141.699 90.48 +134.492 96.41 131.527 c 102.336 128.562 109.543 130.965 112.508 136.895 +c h +112.508 136.895 m S +BT +9.9626 0 0 9.9626 1 190.528 Tm +/f-0-0 1 Tf +[<010203>-333<04>83<050607>28<020608>-333<09020a0a>]TJ +ET +17 142.527 m 89 142.527 l S +89 142.527 m 84 144.191 l 84 140.863 l h +89 142.527 m f* +89 142.527 m 84 144.191 l 84 140.863 l h +89 142.527 m S +BT +9.9626 0 0 9.9626 1 142.531 Tm +/f-1-0 1 Tf +<01>Tj +6.9738 0 0 6.9738 6.694 141.037 Tm +/f-2-0 1 Tf +<01>Tj +9.9626 0 0 9.9626 1 62.531 Tm +/f-1-0 1 Tf +<01>Tj +6.9738 0 0 6.9738 6.694 61.037 Tm +/f-2-0 1 Tf +<02>Tj +9.9626 0 0 9.9626 289 102.528 Tm +/f-1-0 1 Tf +<02>Tj +/f-0-0 1 Tf +[<>-58<01>]TJ +/f-1-0 1 Tf +<03>Tj +6.9738 0 0 6.9738 305.687 101.033 Tm +/f-2-0 1 Tf +<03>Tj +9.9626 0 0 9.9626 310.157 102.528 Tm +/f-1-0 1 Tf +[<04>-167<05>]TJ +/f-0-0 1 Tf +<03>Tj +ET +17 62.527 m 89 62.527 l S +89 62.527 m 84 64.191 l 84 60.863 l h +89 62.527 m f* +89 62.527 m 84 64.191 l 84 60.863 l h +89 62.527 m S +209 102.527 m 281 102.527 l S +281 102.527 m 276 104.191 l 276 100.863 l h +281 102.527 m f* +281 102.527 m 276 104.191 l 276 100.863 l h +281 102.527 m S +112.121 57.195 m 115.086 63.125 112.684 70.332 106.754 73.297 c 100.828 +76.262 93.617 73.855 90.656 67.93 c 87.691 62 90.094 54.793 96.02 +51.828 c 101.949 48.867 109.156 51.27 112.121 57.195 c h +112.121 57.195 m S +207.516 96.852 m 210.48 102.777 208.078 109.988 202.148 112.949 c +196.223 115.914 189.016 113.512 186.051 107.586 c 183.086 101.656 +185.488 94.449 191.418 91.484 c 197.344 88.52 204.551 90.922 207.516 +96.852 c h +207.516 96.852 m S +BT +9.9626 0 0 9.9626 97 62.534 Tm +/f-1-0 1 Tf +<03>Tj +6.9738 0 0 6.9738 101.884 61.04 Tm +/f-2-0 1 Tf +<02>Tj +ET +0.827 g +101.812 34.32 39.926 15.941 re f +0 g +BT +9.9626 0 0 9.9626 104.800078 39.798592 Tm +/f-1-0 1 Tf +[<06>-56<02>-57<0706>-56<03>]TJ +6.9738 0 0 6.9738 134.280078 38.304592 Tm +/f-2-0 1 Tf +<02>Tj +ET +0.827 g +289 1.047 87.062 15.941 re f +0 g +BT +9.9626 0 0 9.9626 291.989 6.526 Tm +/f-0-0 1 Tf +[<010b03>-333<0c020d>27<0e07>28<020608>-333<09020a0a>]TJ +/f-1-0 1 Tf +-25.193122 4.015518 Td +<08>Tj +6.9738 0 0 6.9738 48.132 45.037 Tm +/f-2-0 1 Tf +<04>Tj +ET +0.827 g +40.613 26.668 42.176 15.941 re f +0 g +BT +9.9626 0 0 9.9626 43.601 32.14872 Tm +/f-1-0 1 Tf +[<06>-56<02>-57<0706>-56<08>]TJ +6.9738 0 0 6.9738 75.328 30.65472 Tm +/f-2-0 1 Tf +<04>Tj +9.9626 0 0 9.9626 41 170.531 Tm +/f-1-0 1 Tf +<08>Tj +6.9738 0 0 6.9738 48.132 169.037 Tm +/f-2-0 1 Tf +<01>Tj +ET +0.827 g +40.613 150.668 42.176 15.941 re f +0 g +BT +9.9626 0 0 9.9626 43.601 156.14872 Tm +/f-1-0 1 Tf +[<06>-56<02>-57<0706>-56<08>]TJ +6.9738 0 0 6.9738 75.328 154.65472 Tm +/f-2-0 1 Tf +<01>Tj +9.9626 0 0 9.9626 25 130.531 Tm +/f-1-0 1 Tf +<08>Tj +6.9738 0 0 6.9738 32.132 129.037 Tm +/f-2-0 1 Tf +<02>Tj +ET +114.086 62.16 m 186.211 95.898 l S +186.211 95.898 m 180.977 95.289 l 182.387 92.27 l h +186.211 95.898 m f* +186.211 95.898 m 180.977 95.289 l 182.387 92.27 l h +186.211 95.898 m S +BT +9.9626 0 0 9.9626 25 90.531 Tm +/f-1-0 1 Tf +<08>Tj +6.9738 0 0 6.9738 32.132 89.037 Tm +/f-2-0 1 Tf +<03>Tj +9.9626 0 0 9.9626 97 142.534 Tm +/f-1-0 1 Tf +<03>Tj +6.9738 0 0 6.9738 101.884 141.04 Tm +/f-2-0 1 Tf +<01>Tj +9.9626 0 0 9.9626 193 102.534 Tm +/f-1-0 1 Tf +<03>Tj +6.9738 0 0 6.9738 197.884 101.04 Tm +/f-2-0 1 Tf +<03>Tj +ET +0.827 g +197.406 73.918 39.926 15.941 re f +0 g +BT +9.9626 0 0 9.9626 200.396511 79.395525 Tm +/f-1-0 1 Tf +[<06>-56<02>-57<0706>-56<03>]TJ +6.9738 0 0 6.9738 229.876511 77.901525 Tm +/f-2-0 1 Tf +<03>Tj +9.9626 0 0 9.9626 153 142.531 Tm +/f-1-0 1 Tf +<08>Tj +6.9738 0 0 6.9738 160.132 141.037 Tm +/f-2-0 1 Tf +<05>Tj +9.9626 0 0 9.9626 153 70.531 Tm +/f-1-0 1 Tf +<08>Tj +6.9738 0 0 6.9738 160.132 69.037 Tm +/f-2-0 1 Tf +<06>Tj +ET +0.827 g +152.613 50.668 42.176 15.941 re f +0 g +BT +9.9626 0 0 9.9626 155.6006 56.14872 Tm +/f-1-0 1 Tf +[<06>-56<02>-57<0706>-56<08>]TJ +6.9738 0 0 6.9738 187.3276 54.65472 Tm +/f-2-0 1 Tf +<06>Tj +ET +97 190.527 m 281 190.527 l S +281 190.527 m 274 192.859 l 274 188.195 l h +281 190.527 m f* +281 190.527 m 274 192.859 l 274 188.195 l h +281 190.527 m S +[ 4 4] 0 d +281 6.527 m 97 6.527 l S +97 6.527 m 104 4.195 l 104 8.859 l h +97 6.527 m f* +[] 0.0 d +97 6.527 m 104 4.195 l 104 8.859 l h +97 6.527 m S +0.827 g +288.812 82.32 38.5 15.941 re f +0 g +BT +9.9626 0 0 9.9626 291.799578 87.798592 Tm +/f-1-0 1 Tf +[<06>-56<02>-57<0706>-56<02>]TJ +ET +[ 4 4] 0 d +209.438 98.582 m 281.438 98.582 l S +209.438 98.582 m 214.438 96.918 l 214.438 100.246 l h +209.438 98.582 m f* +[] 0.0 d +209.438 98.582 m 214.438 96.918 l 214.438 100.246 l h +209.438 98.582 m S +[ 4 4] 0 d +114.086 58.16 m 186.211 91.898 l S +114.086 58.16 m 119.32 58.77 l 117.906 61.785 l h +114.086 58.16 m f* +[] 0.0 d +114.086 58.16 m 119.32 58.77 l 117.906 61.785 l h +114.086 58.16 m S +[ 4 4] 0 d +17 58.527 m 89 58.527 l S +17 58.527 m 22 56.863 l 22 60.191 l h +17 58.527 m f* +[] 0.0 d +17 58.527 m 22 56.863 l 22 60.191 l h +17 58.527 m S +[ 4 4] 0 d +113.695 138.047 m 185.438 104.309 l S +113.695 138.047 m 117.512 134.41 l 118.93 137.426 l h +113.695 138.047 m f* +[] 0.0 d +113.695 138.047 m 117.512 134.41 l 118.93 137.426 l h +113.695 138.047 m S +113.695 142.047 m 185.438 108.309 l S +185.438 108.309 m 181.621 111.941 l 180.203 108.93 l h +185.438 108.309 m f* +185.438 108.309 m 181.621 111.941 l 180.203 108.93 l h +185.438 108.309 m S +15.117 137.664 m 91.98 70.691 l S +91.98 70.691 m 89.305 75.23 l 87.117 72.723 l h +91.98 70.691 m f* +91.98 70.691 m 89.305 75.23 l 87.117 72.723 l h +91.98 70.691 m S +15.375 67.957 m 92.754 133.902 l S +92.754 133.902 m 87.871 131.926 l 90.027 129.391 l h +92.754 133.902 m f* +92.754 133.902 m 87.871 131.926 l 90.027 129.391 l h +92.754 133.902 m S +[ 4 4] 0 d +17 138.527 m 89 138.527 l S +17 138.527 m 22 136.863 l 22 140.191 l h +17 138.527 m f* +[] 0.0 d +17 138.527 m 22 136.863 l 22 140.191 l h +17 138.527 m S +[ 4 4] 0 d +15.117 133.664 m 91.98 66.691 l S +15.117 133.664 m 17.793 129.121 l 19.98 131.633 l h +15.117 133.664 m f* +[] 0.0 d +15.117 133.664 m 17.793 129.121 l 19.98 131.633 l h +15.117 133.664 m S +[ 4 4] 0 d +15.375 63.957 m 92.754 129.902 l S +15.375 63.957 m 20.262 65.934 l 18.102 68.469 l h +15.375 63.957 m f* +[] 0.0 d +15.375 63.957 m 20.262 65.934 l 18.102 68.469 l h +15.375 63.957 m S +0.827 g +24.613 70.668 42.176 15.941 re f +0 g +BT +9.9626 0 0 9.9626 27.600643 76.148759 Tm +/f-1-0 1 Tf +[<06>-56<02>-57<0706>-56<08>]TJ +6.9738 0 0 6.9738 59.327643 74.654759 Tm +/f-2-0 1 Tf +<03>Tj +ET +0.827 g +101.461 113.621 39.926 15.941 re f +0 g +BT +9.9626 0 0 9.9626 104.449471 119.098494 Tm +/f-1-0 1 Tf +[<06>-56<02>-57<0706>-56<03>]TJ +6.9738 0 0 6.9738 133.929471 117.604494 Tm +/f-2-0 1 Tf +<01>Tj +ET +0.827 g +152.609 122.672 42.176 15.941 re f +0 g +BT +9.9626 0 0 9.9626 155.600152 128.149072 Tm +/f-1-0 1 Tf +[<06>-56<02>-57<0706>-56<08>]TJ +6.9738 0 0 6.9738 187.327152 126.655072 Tm +/f-2-0 1 Tf +<05>Tj +ET +0.827 g +24.613 110.672 42.176 15.941 re f +0 g +BT +9.9626 0 0 9.9626 27.600803 116.14912 Tm +/f-1-0 1 Tf +[<06>-56<02>-57<0706>-56<08>]TJ +6.9738 0 0 6.9738 59.327803 114.65512 Tm +/f-2-0 1 Tf +<02>Tj +ET +Q +showpage +%%Trailer +count op_count sub {pop} repeat +countdictstack dict_count sub {end} repeat +cairo_eps_state restore +%%EOF diff --git a/doc/Articles/Autodiff/figures/comp-graph/comp-graph.tex b/doc/Articles/Autodiff/figures/comp-graph/comp-graph.tex new file mode 100644 index 000000000..4432fe907 --- /dev/null +++ b/doc/Articles/Autodiff/figures/comp-graph/comp-graph.tex @@ -0,0 +1,43 @@ +\documentclass[tikz]{standalone} +\usepackage{framed} +\usepackage{amsmath} +\usepackage{amsfonts} +\DeclareMathOperator{\f}{f} +\usepackage{xcolor} + +\usetikzlibrary{arrows} +\usetikzlibrary{positioning} + +\begin{document} +\begin{tikzpicture}[] + + \tikzstyle{vnode} = [circle,draw,thick,fill=white,minimum size=9mm] + \tikzstyle{vedge} = [->,>=latex,thick] + + \node[vnode] (v-1) at (-8.5,0.5) {$v_{-1}$}; + \node[vnode] (v0) at (-8.5,-2.5) {$v_0$}; + \node[vnode] (v1) at (-6,0.5) {$v_1$}; + \node[vnode] (v2) at (-6,-1) {$v_2$}; + \node[vnode] (v3) at (-3.5,-2.5) {$v_3$}; + \node[vnode] (v4) at (-3.5,0.5) {$v_4$}; + \node[vnode] (v5) at (-1,-1) {$v_5$}; + + \node[] (x1) at (-11,0.5) {$x_1$}; + \node[] (x2) at (-11,-2.5) {$x_2$}; + \node[] (f) at (1.5,-1) {$f(x_1,x_2)$}; + + \draw (v-1) edge [vedge] (v1); + \draw (v-1) edge [vedge] (v2); + \draw (v0) edge [vedge] (v2); + \draw (v0) edge [vedge] (v3); + \draw (v1) edge [vedge] (v4); + \draw (v2) edge [vedge] (v4); + \draw (v3) edge [vedge] (v5); + \draw (v4) edge [vedge] (v5); + + \draw (x1) edge [vedge] (v-1); + \draw (x2) edge [vedge] (v0); + \draw (v5) edge [vedge] (f); + +\end{tikzpicture} +\end{document} diff --git a/doc/Articles/Autodiff/figures/differentiation/differentiation.tex b/doc/Articles/Autodiff/figures/differentiation/differentiation.tex new file mode 100644 index 000000000..56eaaa9c2 --- /dev/null +++ b/doc/Articles/Autodiff/figures/differentiation/differentiation.tex @@ -0,0 +1,79 @@ +\documentclass[tikz]{standalone} +\usepackage{framed} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{xcolor} + +\usetikzlibrary{decorations.pathmorphing} +\usetikzlibrary{arrows} + +\begin{document} +\begin{tikzpicture}[ + pencildraw/.style={ + decorate, + decoration={random steps,segment length=2pt,amplitude=1pt} + }] + %\draw[help lines] (0,0) grid (20,10); + \tikzstyle{paperbox} = [pencildraw,draw,thick,fill=white,text width=7cm,inner sep=5mm] + \tikzstyle{codebox} = [draw,thick,text width=7cm,inner sep=5mm] + \tikzstyle{edge} = [->,>=triangle 60,thick] + + \node[paperbox] (a) at (-5.5,-12.5) { + $l_1=x$\\ + $l_{n+1}=4l_n(1-l_n)$\\ + \vspace{4mm} + $f(x)=l_4=64x(1 - x)(1 - 2 x)^2 (1 - 8 x + 8 x^2)^2$ + }; + \node[paperbox,fill=gray!20] (b) at (5.5,-12.5) {$f'(x)=128x(1 - x)(-8 + 16 x)(1 - 2 x)^2(1 - 8 x + 8 x^2) + 64 (1 - x)(1 - 2 x)^2 (1 - 8 x + 8 x^2)^2 - 64x(1 - 2 x)^2 (1 - 8 x + 8 x^2)^2 - 256x(1 - x)(1 - 2 x)(1 - 8 x + 8 x^2)^2$}; + \node[codebox,fill=white] (c) at (-5.5,-18.5) {\texttt{\parbox{7cm}{ + f(x):\\ + \hphantom{tt} v = x\\ + \hphantom{tt} for i = 1 to 3\\ + \hphantom{tttt} v = 4*v*(1 - v)\\ + \hphantom{tt} return v\\ + \hphantom{t}\\ + \textrm{or, in closed-form,}\\ + \hphantom{t}\\ + f(x):\\ + \hphantom{tt}\parbox{6cm}{\hangindent=0.4cm \hangafter=1 return 64*x*(1-x)*((1-2*x)\^{}2)\\ *(1-8*x+8*x*x)\^{}2}\\ + }}}; + \node[codebox,black,fill=gray!20] (d) at (5.5,-18.5) {\color{black}\texttt{\parbox{7cm}{\textbf{ + f'(x):\\ + \hphantom{tt}\parbox{6cm}{\hangindent=0.4cm \hangafter=1 return 128*x*(1 - x)*(-8 + 16*x)\\ *((1 - 2*x)\^{}2)*(1 - 8*x + 8*x*x)\\+ 64*(1 - x)*((1 - 2*x)\^{}2)*((1 - 8*x + 8*x*x)\^{}2) - (64*x*(1 - 2*x)\^{}2)*(1 - 8*x + 8*x*x)\^{}2 - 256*x*(1 - x)*(1 - 2*x)*(1 - 8*x + 8*x*x)\^{}2}\\} + \flushright \color{gray} f'($\mathtt{x_0}$) $= f'(x_0)$\\\textrm{Exact} + }}}; + \node[codebox,black,fill=gray!20] (e) at (-5.5,-25.5) {\color{black}\texttt{\parbox{7cm}{\textbf{ + f'(x):\\ + \hphantom{tt} (v,dv) = (x,1)\\ + \hphantom{tt} for i = 1 to 3\\ + \hphantom{tttt} (v,dv) = (4*v*(1-v), 4*dv-8*v*dv)\\ + \hphantom{tt} return (v,dv)\\} + \flushright \color{gray} f'($\mathtt{x_0}$) $= f'(x_0)$\\\textrm{Exact} + }}}; + \node[codebox,black,fill=gray!20] (f) at (5.5,-25.5) {\color{black}\texttt{\parbox{7cm}{\textbf{ + f'(x):\\ + \hphantom{tt} h = 0.000001\\ + \hphantom{tt} return (f(x + h) - f(x)) / h\\} + \flushright \color{gray} f'($\mathtt{x_0}$) $\approx f'(x_0)$\\\textrm{Approximate} + }}}; + + \draw (a) edge [edge] (b); + \node[align=center,below] at (0,-12.5) {Manual\\Differentiation}; + + \draw (c) edge [edge] (d); + \node[align=center,below] at (0,-18.5) {Symbolic\\Differentiation\\of the Closed-form}; + + \draw (a) edge [edge] (c); + \node[align=left,right] at (-5.5,-15) {Coding}; + + \draw (b) edge [edge] (d); + \node[align=left,right] at (5.5,-15) {Coding}; + + \draw (c) edge [edge] (f); + \node[align=left,left] at (0.55,-22.25) {Numerical\\Differentiation}; + + \draw (c) edge [edge] (e); + \node[align=left,right] at (-5.5,-22.25) {Automatic\\Differentiation}; + +\end{tikzpicture} +\end{document} diff --git a/doc/Articles/Autodiff/figures/helmholtz/helmholtz.tex b/doc/Articles/Autodiff/figures/helmholtz/helmholtz.tex new file mode 100644 index 000000000..833dc8861 --- /dev/null +++ b/doc/Articles/Autodiff/figures/helmholtz/helmholtz.tex @@ -0,0 +1,598 @@ +% Created by tikzDevice version 0.8.1 on 2015-02-19 05:31:36 +% !TEX encoding = UTF-8 Unicode +\begin{tikzpicture}[x=1pt,y=1pt] +\definecolor{fillColor}{RGB}{255,255,255} +\path[use as bounding box,fill=fillColor,fill opacity=0.00] (0,0) rectangle (325.21,433.62); +\begin{scope} +\path[clip] ( 54.00,198.30) rectangle (313.21,421.62); +\definecolor{drawColor}{RGB}{0,0,0} + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 63.60,206.61) -- + ( 97.89,206.78) -- + (132.18,207.17) -- + (166.46,207.78) -- + (200.75,208.75) -- + (235.04,210.05) -- + (269.33,211.84) -- + (303.61,213.79); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 61.61,204.62) rectangle ( 65.59,208.61); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 95.89,204.79) rectangle ( 99.88,208.78); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (130.18,205.18) rectangle (134.17,209.17); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (164.47,205.78) rectangle (168.46,209.77); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (198.76,206.75) rectangle (202.75,210.74); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (233.05,208.05) rectangle (237.03,212.04); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (267.33,209.84) rectangle (271.32,213.83); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (301.62,211.79) rectangle (305.61,215.78); +\end{scope} +\begin{scope} +\path[clip] ( 0.00, 0.00) rectangle (325.21,433.62); +\definecolor{drawColor}{RGB}{0,0,0} + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,206.57) -- ( 54.00,413.35); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,206.57) -- ( 48.00,206.57); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,247.93) -- ( 48.00,247.93); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,289.28) -- ( 48.00,289.28); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,330.64) -- ( 48.00,330.64); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,371.99) -- ( 48.00,371.99); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,413.35) -- ( 48.00,413.35); + +\node[text=drawColor,rotate= 90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60,206.57) {0}; + +\node[text=drawColor,rotate= 90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60,247.93) {1000}; + +\node[text=drawColor,rotate= 90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60,289.28) {2000}; + +\node[text=drawColor,rotate= 90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60,330.64) {3000}; + +\node[text=drawColor,rotate= 90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60,371.99) {4000}; + +\node[text=drawColor,rotate= 90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60,413.35) {5000}; + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,198.30) -- + (313.21,198.30) -- + (313.21,421.62) -- + ( 54.00,421.62) -- + ( 54.00,198.30); +\end{scope} +\begin{scope} +\path[clip] ( 54.00,198.30) rectangle (313.21,421.62); +\definecolor{drawColor}{RGB}{0,0,0} + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 4pt off 4pt ,line join=round,line cap=round] ( 63.60,206.62) -- + ( 97.89,208.04) -- + (132.18,213.88) -- + (166.46,227.23) -- + (200.75,249.80) -- + (235.04,288.73) -- + (269.33,341.78) -- + (303.61,413.18); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 63.60,203.80) -- + ( 66.42,206.62) -- + ( 63.60,209.44) -- + ( 60.78,206.62) -- + ( 63.60,203.80); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 97.89,205.22) -- + (100.71,208.04) -- + ( 97.89,210.86) -- + ( 95.07,208.04) -- + ( 97.89,205.22); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (132.18,211.06) -- + (135.00,213.88) -- + (132.18,216.70) -- + (129.36,213.88) -- + (132.18,211.06); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (166.46,224.41) -- + (169.28,227.23) -- + (166.46,230.05) -- + (163.64,227.23) -- + (166.46,224.41); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (200.75,246.98) -- + (203.57,249.80) -- + (200.75,252.62) -- + (197.93,249.80) -- + (200.75,246.98); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (235.04,285.91) -- + (237.86,288.73) -- + (235.04,291.55) -- + (232.22,288.73) -- + (235.04,285.91); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (269.33,338.96) -- + (272.15,341.78) -- + (269.33,344.60) -- + (266.51,341.78) -- + (269.33,338.96); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (303.61,410.36) -- + (306.43,413.18) -- + (303.61,416.00) -- + (300.79,413.18) -- + (303.61,410.36); + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt ,line join=round,line cap=round] ( 63.60,206.63) -- + ( 97.89,207.14) -- + (132.18,208.70) -- + (166.46,212.04) -- + (200.75,216.96) -- + (235.04,226.00) -- + (269.33,240.30) -- + (303.61,262.07); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 63.60,210.13) -- + ( 66.63,204.88) -- + ( 60.57,204.88) -- + ( 63.60,210.13); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 97.89,210.64) -- + (100.92,205.39) -- + ( 94.86,205.39) -- + ( 97.89,210.64); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (132.18,212.20) -- + (135.21,206.95) -- + (129.15,206.95) -- + (132.18,212.20); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (166.46,215.54) -- + (169.49,210.29) -- + (163.43,210.29) -- + (166.46,215.54); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (200.75,220.46) -- + (203.78,215.22) -- + (197.72,215.22) -- + (200.75,220.46); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (235.04,229.50) -- + (238.07,224.25) -- + (232.01,224.25) -- + (235.04,229.50); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (269.33,243.80) -- + (272.36,238.55) -- + (266.30,238.55) -- + (269.33,243.80); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (303.61,265.57) -- + (306.64,260.32) -- + (300.58,260.32) -- + (303.61,265.57); + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt on 4pt off 3pt ,line join=round,line cap=round] ( 63.60,206.63) -- + ( 97.89,207.03) -- + (132.18,207.87) -- + (166.46,209.35) -- + (200.75,211.29) -- + (235.04,213.79) -- + (269.33,217.08) -- + (303.61,220.73); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 63.60,203.14) -- + ( 66.63,208.38) -- + ( 60.57,208.38) -- + ( 63.60,203.14); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 97.89,203.53) -- + (100.92,208.78) -- + ( 94.86,208.78) -- + ( 97.89,203.53); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (132.18,204.37) -- + (135.21,209.62) -- + (129.15,209.62) -- + (132.18,204.37); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (166.46,205.85) -- + (169.49,211.10) -- + (163.43,211.10) -- + (166.46,205.85); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (200.75,207.79) -- + (203.78,213.03) -- + (197.72,213.03) -- + (200.75,207.79); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (235.04,210.29) -- + (238.07,215.54) -- + (232.01,215.54) -- + (235.04,210.29); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (269.33,213.58) -- + (272.36,218.83) -- + (266.30,218.83) -- + (269.33,213.58); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (303.61,217.23) -- + (306.64,222.48) -- + (300.58,222.48) -- + (303.61,217.23); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 63.60,411.28) rectangle (207.00,351.28); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 66.30,399.28) -- ( 84.30,399.28); + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 4pt off 4pt ,line join=round,line cap=round] ( 66.30,387.28) -- ( 84.30,387.28); + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt ,line join=round,line cap=round] ( 66.30,375.28) -- ( 84.30,375.28); + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt on 4pt off 3pt ,line join=round,line cap=round] ( 66.30,363.28) -- ( 84.30,363.28); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 73.31,397.29) rectangle ( 77.29,401.28); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 75.30,384.46) -- + ( 78.12,387.28) -- + ( 75.30,390.10) -- + ( 72.48,387.28) -- + ( 75.30,384.46); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 75.30,378.78) -- + ( 78.33,373.53) -- + ( 72.27,373.53) -- + ( 75.30,378.78); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 75.30,359.78) -- + ( 78.33,365.03) -- + ( 72.27,365.03) -- + ( 75.30,359.78); + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 93.30,395.84) {$f$, original function}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 93.30,383.84) {$\nabla f$, numerical diff.}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 93.30,371.84) {$\nabla f$, forward AD}; + +\node[text=drawColor,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 93.30,359.84) {$\nabla f$, reverse AD}; +\end{scope} +\begin{scope} +\path[clip] ( 54.00, 54.00) rectangle (313.21,185.10); +\definecolor{drawColor}{RGB}{0,0,0} + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 63.60, 58.86) -- + ( 97.89, 82.16) -- + (132.18, 96.98) -- + (166.46,106.90) -- + (200.75,115.33) -- + (235.04,122.01) -- + (269.33,127.93) -- + (303.61,132.42); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 61.61, 56.86) rectangle ( 65.59, 60.85); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 95.89, 80.16) rectangle ( 99.88, 84.15); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (130.18, 94.99) rectangle (134.17, 98.98); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (164.47,104.91) rectangle (168.46,108.90); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (198.76,113.34) rectangle (202.75,117.32); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (233.05,120.01) rectangle (237.03,124.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (267.33,125.94) rectangle (271.32,129.93); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (301.62,130.43) rectangle (305.61,134.41); +\end{scope} +\begin{scope} +\path[clip] ( 0.00, 0.00) rectangle (325.21,433.62); +\definecolor{drawColor}{RGB}{0,0,0} + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 58.70, 54.00) -- (303.61, 54.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 58.70, 54.00) -- ( 58.70, 48.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (107.68, 54.00) -- (107.68, 48.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (156.67, 54.00) -- (156.67, 48.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (205.65, 54.00) -- (205.65, 48.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (254.63, 54.00) -- (254.63, 48.00); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (303.61, 54.00) -- (303.61, 48.00); + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 58.70, 32.40) {0}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at (107.68, 32.40) {10}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at (156.67, 32.40) {20}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at (205.65, 32.40) {30}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at (254.63, 32.40) {40}; + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at (303.61, 32.40) {50}; + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 54.00) -- + (313.21, 54.00) -- + (313.21,185.10) -- + ( 54.00,185.10) -- + ( 54.00, 54.00); +\end{scope} +\begin{scope} +\path[clip] ( 0.00, 0.00) rectangle (325.21,197.10); +\definecolor{drawColor}{RGB}{0,0,0} + +\node[text=drawColor,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at (183.61, 8.40) {$n$}; +\end{scope} +\begin{scope} +\path[clip] ( 0.00, 0.00) rectangle (325.21,433.62); +\definecolor{drawColor}{RGB}{0,0,0} + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 58.86) -- ( 54.00,157.31); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 58.86) -- ( 48.00, 58.86); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 91.67) -- ( 48.00, 91.67); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,124.49) -- ( 48.00,124.49); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,157.31) -- ( 48.00,157.31); + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60, 52.11) {10}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at ( 35.51, 62.10) {0}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60, 84.92) {10}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at ( 35.51, 94.92) {1}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60,117.74) {10}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at ( 35.51,127.74) {2}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 39.60,150.56) {10}; + +\node[text=drawColor,rotate= 90.00,anchor=base west,inner sep=0pt, outer sep=0pt, scale= 0.70] at ( 35.51,160.56) {3}; + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 55.68) -- ( 54.00,185.04); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 55.68) -- ( 50.40, 55.68); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 57.35) -- ( 50.40, 57.35); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 68.73) -- ( 50.40, 68.73); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 74.51) -- ( 50.40, 74.51); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 78.61) -- ( 50.40, 78.61); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 81.79) -- ( 50.40, 81.79); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 84.39) -- ( 50.40, 84.39); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 86.59) -- ( 50.40, 86.59); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 88.49) -- ( 50.40, 88.49); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00, 90.17) -- ( 50.40, 90.17); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,101.55) -- ( 50.40,101.55); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,107.33) -- ( 50.40,107.33); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,111.43) -- ( 50.40,111.43); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,114.61) -- ( 50.40,114.61); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,117.21) -- ( 50.40,117.21); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,119.41) -- ( 50.40,119.41); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,121.31) -- ( 50.40,121.31); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,122.99) -- ( 50.40,122.99); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,134.37) -- ( 50.40,134.37); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,140.15) -- ( 50.40,140.15); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,144.25) -- ( 50.40,144.25); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,147.43) -- ( 50.40,147.43); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,150.03) -- ( 50.40,150.03); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,152.22) -- ( 50.40,152.22); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,154.13) -- ( 50.40,154.13); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,155.80) -- ( 50.40,155.80); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,167.19) -- ( 50.40,167.19); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,172.96) -- ( 50.40,172.96); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,177.06) -- ( 50.40,177.06); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,180.24) -- ( 50.40,180.24); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,182.84) -- ( 50.40,182.84); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 54.00,185.04) -- ( 50.40,185.04); +\end{scope} +\begin{scope} +\path[clip] ( 54.00, 54.00) rectangle (313.21,185.10); +\definecolor{drawColor}{RGB}{0,0,0} + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 4pt off 4pt ,line join=round,line cap=round] ( 63.60, 60.04) -- + ( 97.89,109.75) -- + (132.18,132.61) -- + (166.46,147.41) -- + (200.75,157.94) -- + (235.04,167.09) -- + (269.33,174.19) -- + (303.61,180.23); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 63.60, 57.22) -- + ( 66.42, 60.04) -- + ( 63.60, 62.86) -- + ( 60.78, 60.04) -- + ( 63.60, 57.22); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 97.89,106.93) -- + (100.71,109.75) -- + ( 97.89,112.57) -- + ( 95.07,109.75) -- + ( 97.89,106.93); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (132.18,129.79) -- + (135.00,132.61) -- + (132.18,135.43) -- + (129.36,132.61) -- + (132.18,129.79); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (166.46,144.59) -- + (169.28,147.41) -- + (166.46,150.23) -- + (163.64,147.41) -- + (166.46,144.59); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (200.75,155.12) -- + (203.57,157.94) -- + (200.75,160.76) -- + (197.93,157.94) -- + (200.75,155.12); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (235.04,164.27) -- + (237.86,167.09) -- + (235.04,169.91) -- + (232.22,167.09) -- + (235.04,164.27); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (269.33,171.37) -- + (272.15,174.19) -- + (269.33,177.01) -- + (266.51,174.19) -- + (269.33,171.37); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (303.61,177.41) -- + (306.43,180.23) -- + (303.61,183.05) -- + (300.79,180.23) -- + (303.61,177.41); + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt ,line join=round,line cap=round] ( 63.60, 63.11) -- + ( 97.89, 96.15) -- + (132.18,115.04) -- + (166.46,128.48) -- + (200.75,137.62) -- + (235.04,146.54) -- + (269.33,154.40) -- + (303.61,161.50); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 63.60, 66.61) -- + ( 66.63, 61.36) -- + ( 60.57, 61.36) -- + ( 63.60, 66.61); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 97.89, 99.65) -- + (100.92, 94.40) -- + ( 94.86, 94.40) -- + ( 97.89, 99.65); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (132.18,118.54) -- + (135.21,113.29) -- + (129.15,113.29) -- + (132.18,118.54); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (166.46,131.98) -- + (169.49,126.73) -- + (163.43,126.73) -- + (166.46,131.98); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (200.75,141.12) -- + (203.78,135.87) -- + (197.72,135.87) -- + (200.75,141.12); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (235.04,150.04) -- + (238.07,144.79) -- + (232.01,144.79) -- + (235.04,150.04); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (269.33,157.90) -- + (272.36,152.65) -- + (266.30,152.65) -- + (269.33,157.90); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (303.61,165.00) -- + (306.64,159.75) -- + (300.58,159.75) -- + (303.61,165.00); + +\path[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt on 4pt off 3pt ,line join=round,line cap=round] ( 63.60, 64.84) -- + ( 97.89, 93.19) -- + (132.18,107.97) -- + (166.46,118.84) -- + (200.75,126.36) -- + (235.04,132.43) -- + (269.33,137.78) -- + (303.61,142.03); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 63.60, 61.34) -- + ( 66.63, 66.59) -- + ( 60.57, 66.59) -- + ( 63.60, 61.34); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 97.89, 89.69) -- + (100.92, 94.94) -- + ( 94.86, 94.94) -- + ( 97.89, 89.69); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (132.18,104.47) -- + (135.21,109.72) -- + (129.15,109.72) -- + (132.18,104.47); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (166.46,115.34) -- + (169.49,120.59) -- + (163.43,120.59) -- + (166.46,115.34); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (200.75,122.86) -- + (203.78,128.11) -- + (197.72,128.11) -- + (200.75,122.86); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (235.04,128.94) -- + (238.07,134.18) -- + (232.01,134.18) -- + (235.04,128.94); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (269.33,134.28) -- + (272.36,139.53) -- + (266.30,139.53) -- + (269.33,134.28); + +\path[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (303.61,138.53) -- + (306.64,143.78) -- + (300.58,143.78) -- + (303.61,138.53); +\end{scope} +\begin{scope} +\path[clip] ( 0.00, 0.00) rectangle (325.21,433.62); +\definecolor{drawColor}{RGB}{0,0,0} + +\node[text=drawColor,rotate= 90.00,anchor=base,inner sep=0pt, outer sep=0pt, scale= 1.00] at ( 15.60,219.76) {Time}; +\end{scope} +\end{tikzpicture} diff --git a/doc/Articles/Autodiff/jmlr2e_mod.sty b/doc/Articles/Autodiff/jmlr2e_mod.sty new file mode 100644 index 000000000..54fd24083 --- /dev/null +++ b/doc/Articles/Autodiff/jmlr2e_mod.sty @@ -0,0 +1,353 @@ +% +% File: Macros for Journal of Machine Learning Research +% Very minor modification of macros for Journal of Artificial +% Intelligence Research (jair.sty) +% +% Suggestions: Submit an issue or pull request to +% https://github.com/JournalMLR/jmlr-style-file +% +% Last edited October 9, 2000 by Leslie Pack Kaelbling +% Last edited January 23, 2001 by Alex J. Smola (we should set up RCS or CVS) +% Last edited March 29, 2004 Erik G. Learned-Miller +% Last edited January 17, 2016 Charles Sutton +% Last edited January 9, 2017 Charles Sutton +% (We have now set up GIT, good thing that we waited for it to +% be invented.) +% +% The name of this file should follow the article document +% type, e.g. \documentstyle[jmlr]{article} + +% Copied and edited from similar file for Machine Learning Journal. +% Original Author: Jeff Schlimmer +% Edited by: Kevin Thompson, Martha Del Alto, Helen Stewart, Steve Minton \& Pandu Nayak. +% Last edited: Mon May 3 20:40:00 1993 by kthompso (Kevin Thompson) on muir + +\typeout{Document Style `jmlr' -- January 2016.} + +\newif\if@abbrvbib\@abbrvbibfalse +\DeclareOption{abbrvbib}{\@abbrvbibtrue} + +\newif\if@usehyper\@usehypertrue +\DeclareOption{nohyperref}{\@usehyperfalse} +\DeclareOption{hyperref}{\@usehypertrue} + +\DeclareOption*{\PackageWarning{jmlr}{Unknown ‘\CurrentOption’}} +\ProcessOptions\relax + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% REQUIRED PACKAGES +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\RequirePackage{epsfig} +\RequirePackage{amssymb} +\RequirePackage{natbib} +\RequirePackage{graphicx} + +\if@usehyper +\RequirePackage[colorlinks=false,allbordercolors={1 1 1}]{hyperref} +\fi + +\if@abbrvbib +\bibliographystyle{abbrvnat} +\else +\bibliographystyle{plainnat} +\fi + +\bibpunct{(}{)}{;}{a}{,}{,} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% P A G E S I Z E +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Change the overall width of the page. If these parameters are +% changed, they will require corresponding changes in the +% maketitle section. +% +\renewcommand{\topfraction}{0.95} % let figure take up nearly whole page +\renewcommand{\textfraction}{0.05} % let figure take up nearly whole page + +% Specify the dimensions of each page + +\oddsidemargin .25in % Note \oddsidemargin = \evensidemargin +\evensidemargin .25in +\marginparwidth 0.07 true in +%\marginparwidth 0.75 true in +%\topmargin 0 true pt % Nominal distance from top of page to top of +%\topmargin 0.125in +\topmargin -0.5in +\addtolength{\headsep}{0.25in} +\textheight 8.5 true in % Height of text (including footnotes & figures) +\textwidth 6.0 true in % Width of text line. +\widowpenalty=10000 +\clubpenalty=10000 +\@twosidetrue \@mparswitchtrue \def\ds@draft{\overfullrule 5pt} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% S E C T I O N S +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Definitions for nicer (?) sections, etc., ideas from Pat Langley. +% Numbering for sections, etc. is taken care of automatically. + +\def\@startsiction#1#2#3#4#5#6{\if@noskipsec \leavevmode \fi + \par \@tempskipa #4\relax + \@afterindenttrue + \ifdim \@tempskipa <\z@ \@tempskipa -\@tempskipa \@afterindentfalse\fi + \if@nobreak \everypar{}\else + \addpenalty{\@secpenalty}\addvspace{\@tempskipa}\fi \@ifstar + {\@ssect{#3}{#4}{#5}{#6}}{\@dblarg{\@sict{#1}{#2}{#3}{#4}{#5}{#6}}}} + +\def\@sict#1#2#3#4#5#6[#7]#8{\ifnum #2>\c@secnumdepth + \def\@svsec{}\else + \refstepcounter{#1}\edef\@svsec{\csname the#1\endcsname}\fi + \@tempskipa #5\relax + \ifdim \@tempskipa>\z@ + \begingroup #6\relax + \@hangfrom{\hskip #3\relax\@svsec.\hskip 0.1em} + {\interlinepenalty \@M #8\par} + \endgroup + \csname #1mark\endcsname{#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}\else + \def\@svsechd{#6\hskip #3\@svsec #8\csname #1mark\endcsname + {#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}}\fi + \@xsect{#5}} + +\def\@sect#1#2#3#4#5#6[#7]#8{\ifnum #2>\c@secnumdepth + \def\@svsec{}\else + \refstepcounter{#1}\edef\@svsec{\csname the#1\endcsname\hskip 0.5em }\fi + \@tempskipa #5\relax + \ifdim \@tempskipa>\z@ + \begingroup #6\relax + \@hangfrom{\hskip #3\relax\@svsec}{\interlinepenalty \@M #8\par} + \endgroup + \csname #1mark\endcsname{#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}\else + \def\@svsechd{#6\hskip #3\@svsec #8\csname #1mark\endcsname + {#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}}\fi + \@xsect{#5}} + +\def\thesection {\arabic{section}} +\def\thesubsection {\thesection.\arabic{subsection}} +\def\section{\@startsiction{section}{1}{\z@}{-0.24in}{0.10in} + {\large\bf\raggedright}} +\def\subsection{\@startsection{subsection}{2}{\z@}{-0.20in}{0.08in} + {\normalsize\bf\raggedright}} +\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-0.18in}{0.08in} + {\normalsize\sc\raggedright}} +\def\paragraph{\@startsiction{paragraph}{4}{\z@}{1.5ex plus + 0.5ex minus .2ex}{-1em}{\normalsize\bf}} +\def\subparagraph{\@startsiction{subparagraph}{5}{\z@}{1.5ex plus + 0.5ex minus .2ex}{-1em}{\normalsize\bf}} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% F O O T N O T E S +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Change the size of the footnote rule +% +% \renewcommand{\footnoterule}{\vspace{10pt}\hrule width 0mm} + +\long\def\@makefntext#1{\@setpar{\@@par\@tempdima \hsize + \advance\@tempdima-15pt\parshape \@ne 15pt \@tempdima}\par + \parindent 2em\noindent \hbox to \z@{\hss{\@thefnmark}. \hfil}#1} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% A B S T R A C T +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%% use \begin{abstract} .. \end{abstract} for abstracts. +\renewenvironment{abstract} +{\centerline{\large\bf Abstract}\vspace{0.7ex}% + \bgroup\leftskip 20pt\rightskip 20pt\small\noindent\ignorespaces}% +{\par\egroup\vskip 0.25ex} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% KEYWORDS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%% use \begin{keywords} .. \end{keywords} for keywordss. +\newenvironment{keywords} +{\bgroup\leftskip 20pt\rightskip 20pt \small\noindent{\bf Keywords:} }% +{\par\egroup\vskip 0.25ex} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% FIRST PAGE, TITLE, AUTHOR +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Author information can be set in various styles: +% For several authors from the same institution: +% \author{Author 1 \and ... \and Author n \\ +% \addr{Address line} \\ ... \\ \addr{Address line}} +% if the names do not fit well on one line use +% Author 1 \\ {\bf Author 2} \\ ... \\ {\bf Author n} \\ +% To start a seperate ``row'' of authors use \AND, as in +% \author{Author 1 \\ \addr{Address line} \\ ... \\ \addr{Address line} +% \AND +% Author 2 \\ \addr{Address line} \\ ... \\ \addr{Address line} \And +% Author 3 \\ \addr{Address line} \\ ... \\ \addr{Address line}} + +% Title stuff, borrowed in part from aaai92.sty + +\newlength\aftertitskip \newlength\beforetitskip +\newlength\interauthorskip \newlength\aftermaketitskip + +%% Changeable parameters. +\setlength\aftertitskip{0.1in plus 0.2in minus 0.2in} +\setlength\beforetitskip{0.05in plus 0.08in minus 0.08in} +\setlength\interauthorskip{0.08in plus 0.1in minus 0.1in} +\setlength\aftermaketitskip{0.3in plus 0.1in minus 0.1in} + +%% overall definition of maketitle, @maketitle does the real work +\def\maketitle{\par + \begingroup + \def\thefootnote{\fnsymbol{footnote}} + \def\@makefnmark{\hbox to 0pt{$^{\@thefnmark}$\hss}} + \@maketitle \@thanks + \endgroup +\setcounter{footnote}{0} + \let\maketitle\relax \let\@maketitle\relax + \gdef\@thanks{}\gdef\@author{}\gdef\@title{}\let\thanks\relax} + +\def\@startauthor{\noindent \normalsize\bf} +\def\@endauthor{} +\def\@starteditor{\noindent \small {\bf ~}} +\def\@endeditor{\normalsize} +\def\@maketitle{\vbox{\hsize\textwidth + \linewidth\hsize \vskip \beforetitskip + {\begin{center} \Large\bf \@title \par \end{center}} \vskip \aftertitskip + {\def\and{\unskip\enspace{\rm and}\enspace}% + \def\addr{\small\it}% + \def\email{\hfill\small\sc}% + \def\name{\normalsize\bf}% + \def\AND{\@endauthor\rm\hss \vskip \interauthorskip \@startauthor} + \@startauthor \@author \@endauthor} + + \vskip \aftermaketitskip + \noindent \@starteditor \@editor \@endeditor + \vskip \aftermaketitskip +}} + +\def\editor#1{\gdef\@editor{}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%% +%%% Pagestyle +%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%% Defines the pagestyle for the title page. +%% Usage: \jmlrheading{1}{1993}{1-15}{8/93}{9/93}{14-115}{Jane Q. Public and A. U. Thor} +%% \jmlrheading{vol}{year}{pages}{Submitted date}{published date}{paper id}{authors} +%% +%% If your paper required revisions that were reviewed by the action editor, then indicate +%% this by, e.g. +%% \jmlrheading{1}{1993}{1-15}{8/93; Revised 10/93}{12/93}{14-115}{Jane Q. Public and A. U. Thor} + +\def\firstpageno#1{\setcounter{page}{#1}} + +\def\jmlrheading#1#2#3#4#5#6#7{\def\ps@jmlrtps{\let\@mkboth\@gobbletwo% +\def\@oddhead{\scriptsize }% +\def\@oddfoot{\parbox[t]{\textwidth}{\raggedright \scriptsize \hfill}}% + +\def\@evenhead{}\def\@evenfoot{}}% +\thispagestyle{jmlrtps}} + +%% Defines the pagestyle for the rest of the pages +%% Usage: \ShortHeadings{Minimizing Conflicts}{Minton et al} +%% \ShortHeadings{short title}{short authors} + +\def\ShortHeadings#1#2{\def\ps@jmlrps{\let\@mkboth\@gobbletwo% +\def\@oddhead{\hfill {\small\sc #1} \hfill}% +\def\@oddfoot{\hfill \small\rm \thepage \hfill}% +\def\@evenhead{\hfill {\small\sc #2} \hfill}% +\def\@evenfoot{\hfill \small\rm \thepage \hfill}}% +\pagestyle{jmlrps}} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% MISCELLANY +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% Define macros for figure captions and table titles + +% Figurecaption prints the caption title flush left. +% \def\figurecaption#1#2{\noindent\hangindent 42pt +% \hbox to 36pt {\sl #1 \hfil} +% \ignorespaces #2} +% \def\figurecaption#1#2{\noindent\hangindent 46pt +% \hbox to 41pt {\small\sl #1 \hfil} +% \ignorespaces {\small #2}} +\def\figurecaption#1#2{\noindent\hangindent 40pt + \hbox to 36pt {\small\sl #1 \hfil} + \ignorespaces {\small #2}} +% Figurecenter prints the caption title centered. +\def\figurecenter#1#2{\centerline{{\sl #1} #2}} +\def\figurecenter#1#2{\centerline{{\small\sl #1} {\small #2}}} + +% +% Allow ``hanging indents'' in long captions +% +\long\def\@makecaption#1#2{ + \vskip 10pt + \setbox\@tempboxa\hbox{#1: #2} + \ifdim \wd\@tempboxa >\hsize % IF longer than one line: + \begin{list}{#1:}{ + \settowidth{\labelwidth}{#1:} + \setlength{\leftmargin}{\labelwidth} + \addtolength{\leftmargin}{\labelsep} + }\item #2 \end{list}\par % Output in quote mode + \else % ELSE center. + \hbox to\hsize{\hfil\box\@tempboxa\hfil} + \fi} + + +% Define strut macros for skipping spaces above and below text in a +% tabular environment. +\def\abovestrut#1{\rule[0in]{0in}{#1}\ignorespaces} +\def\belowstrut#1{\rule[-#1]{0in}{#1}\ignorespaces} + +% Acknowledgments +\long\def\acks#1{\vskip 0.3in\noindent{\large\bf Acknowledgments}\vskip 0.2in +\noindent #1} + +% Research Note +\long\def\researchnote#1{\noindent {\LARGE\it Research Note} #1} + +\renewcommand{\appendix}{\par + \setcounter{section}{0} + \setcounter{subsection}{0} + \def\thesection{\Alph{section}} +\def\section{\@ifnextchar*{\@startsiction{section}{1}{\z@}{-0.24in}{0.10in}% + {\large\bf\raggedright}}% +{\@startsiction{section}{1}{\z@}{-0.24in}{0.10in} + {\large\bf\raggedright Appendix\ }}}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% PROOF, THEOREM, and FRIENDS +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newcommand{\BlackBox}{\rule{1.5ex}{1.5ex}} % end of proof +\newenvironment{proof}{\par\noindent{\bf Proof\ }}{\hfill\BlackBox\\[2mm]} +\newtheorem{example}{Example} +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{conjecture}[theorem]{Conjecture} +\newtheorem{axiom}[theorem]{Axiom} + + + diff --git a/doc/Articles/CNN.pdf b/doc/Articles/CNN.pdf new file mode 100644 index 000000000..cae8d4b6c Binary files /dev/null and b/doc/Articles/CNN.pdf differ diff --git a/doc/Articles/Fig1.pdf b/doc/Articles/Fig1.pdf new file mode 100644 index 000000000..79e0d66bd Binary files /dev/null and b/doc/Articles/Fig1.pdf differ diff --git a/doc/Articles/bibliography.bib b/doc/Articles/bibliography.bib new file mode 100644 index 000000000..8f7c46955 --- /dev/null +++ b/doc/Articles/bibliography.bib @@ -0,0 +1,3268 @@ +%% This BibTeX bibliography file was created using BibDesk. +%% https://bibdesk.sourceforge.io/ + +%% Created for Giacomo Torlai at 2020-05-26 12:26:10 -0400 + + +%% Saved with string encoding Unicode (UTF-8) + +@misc{coderepo, + title={https://github.com/GTorlai/NeuralNetworks-for-Quantum}, + url={https://github.com/GTorlai/NeuralNetworks-for-Quantum} +} + +@ARTICLE{2012arXiv1212.5701Z, + author = {{Zeiler}, Matthew D.}, + title = "{ADADELTA: An Adaptive Learning Rate Method}", + journal = {arXiv e-prints}, + keywords = {Computer Science - Machine Learning}, + year = 2012, + month = dec, + eid = {arXiv:1212.5701}, + pages = {arXiv:1212.5701}, +archivePrefix = {arXiv}, + eprint = {1212.5701}, + primaryClass = {cs.LG}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2012arXiv1212.5701Z}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@ARTICLE{2020arXiv201212281E, + author = {{Ebadi}, Sepehr and {Wang}, Tout T. and {Levine}, Harry and {Keesling}, Alexander and {Semeghini}, Giulia and {Omran}, Ahmed and {Bluvstein}, Dolev and {Samajdar}, Rhine and {Pichler}, Hannes and {Ho}, Wen Wei and {Choi}, Soonwon and {Sachdev}, Subir and {Greiner}, Markus and {Vuletic}, Vladan and {Lukin}, Mikhail D.}, + title = "{Quantum Phases of Matter on a 256-Atom Programmable Quantum Simulator}", + journal = {arXiv e-prints}, + keywords = {Quantum Physics, Condensed Matter - Quantum Gases, Physics - Atomic Physics}, + year = 2020, + month = dec, + eid = {arXiv:2012.12281}, + pages = {arXiv:2012.12281}, +archivePrefix = {arXiv}, + eprint = {2012.12281}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv201212281E}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@ARTICLE{2020arXiv201212268S, + author = {{Scholl}, Pascal and {Schuler}, Michael and {Williams}, Hannah J. and {Eberharter}, Alexander A. and {Barredo}, Daniel and {Schymik}, Kai-Niklas and {Lienhard}, Vincent and {Henry}, Louis-Paul and {Lang}, Thomas C. and {Lahaye}, Thierry and {L{\"a}uchli}, Andreas M. and {Browaeys}, Antoine}, + title = "{Programmable quantum simulation of 2D antiferromagnets with hundreds of Rydberg atoms}", + journal = {arXiv e-prints}, + keywords = {Quantum Physics, Condensed Matter - Quantum Gases, Physics - Atomic Physics}, + year = 2020, + month = dec, + eid = {arXiv:2012.12268}, + pages = {arXiv:2012.12268}, +archivePrefix = {arXiv}, + eprint = {2012.12268}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv201212268S}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@article{mnih_human-level_2015, + title = {Human-level control through deep reinforcement learning}, + volume = {518}, + issn = {1476-4687}, + url = {https://doi.org/10.1038/nature14236}, + doi = {10.1038/nature14236}, + abstract = {An artificial agent is developed that learns to play a diverse range of classic Atari 2600 computer games directly from sensory experience, achieving a performance comparable to that of an expert human player; this work paves the way to building general-purpose learning algorithms that bridge the divide between perception and action.}, + number = {7540}, + journal = {Nature}, + author = {Mnih, Volodymyr and Kavukcuoglu, Koray and Silver, David and Rusu, Andrei A. and Veness, Joel and Bellemare, Marc G. and Graves, Alex and Riedmiller, Martin and Fidjeland, Andreas K. and Ostrovski, Georg and Petersen, Stig and Beattie, Charles and Sadik, Amir and Antonoglou, Ioannis and King, Helen and Kumaran, Dharshan and Wierstra, Daan and Legg, Shane and Hassabis, Demis}, + month = feb, + year = {2015}, + pages = {529--533}, +} + + +@book{becca_sorella_2017, place={Cambridge}, title={Quantum Monte Carlo Approaches for Correlated Systems}, DOI={10.1017/9781316417041}, publisher={Cambridge University Press}, author={Becca, Federico and Sorella, Sandro}, year={2017}} + + +@ARTICLE{2014arXiv1412.6980K, + author = {{Kingma}, Diederik P. and {Ba}, Jimmy}, + title = "{Adam: A Method for Stochastic Optimization}", + journal = {arXiv e-prints}, + keywords = {Computer Science - Machine Learning}, + year = 2014, + month = dec, + eid = {arXiv:1412.6980}, + pages = {arXiv:1412.6980}, +archivePrefix = {arXiv}, + eprint = {1412.6980}, + primaryClass = {cs.LG}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2014arXiv1412.6980K}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + + +@article{haffner_scalable_2005, + title = {Scalable multiparticle entanglement of trapped ions}, + volume = {438}, + issn = {1476-4687}, + url = {https://doi.org/10.1038/nature04279}, + doi = {10.1038/nature04279}, + abstract = {Schrödinger's hypothetical cat was both dead and alive thanks to a paradox of quantum mechanics, in which a system exists in two or more states at once in a ‘superposition’ of entangled states. Creating this situation experimentally is very difficult, especially for systems made up of many particles, as interactions with the environment destroy superposition in a process called decoherence. So far, entangled states of just a handful of atoms or photons have been achieved. Now, two groups have extended the limits of quantum state engineering by creating the largest entangled atomic systems to date. Working with atoms held in an ion trap by an electromagnetic field to limit decoherence, a team from the National Institute of Standards and Technology in Boulder, Colorado, has created ‘cat states’ of up to six beryllium atoms. A second group, based at Innsbruck University in Austria, has achieved a similar feat by making a related entangled state, a ‘W state’, containing up to eight particles. As the states are created ‘on demand’ and should be scalable to many more particles, there is hope that this technology will pave the way for building a large-scale quantum computer.}, + number = {7068}, + journal = {Nature}, + author = {Häffner, H. and Hänsel, W. and Roos, C. F. and Benhelm, J. and Chek-al-kar, D. and Chwalla, M. and Körber, T. and Rapol, U. D. and Riebe, M. and Schmidt, P. O. and Becher, C. and Gühne, O. and Dür, W. and Blatt, R.}, + month = dec, + year = {2005}, + pages = {643--646}, +} + +@ARTICLE{Bravyi2006, + author = {{Bravyi}, Sergey and {DiVincenzo}, David P. and {Oliveira}, Roberto I. and {Terhal}, Barbara M.}, + title = "{The Complexity of Stoquastic Local Hamiltonian Problems}", + journal = {arXiv e-prints}, + keywords = {Quantum Physics}, + year = 2006, + month = jun, + eid = {quant-ph/0606140}, + pages = {quant-ph/0606140}, +archivePrefix = {arXiv}, + eprint = {quant-ph/0606140}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2006quant.ph..6140B}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@article{LeRoux2008, +author = {Le Roux, Nicolas and Bengio, Yoshua}, +title = {Representational Power of Restricted Boltzmann Machines and Deep Belief Networks}, +journal = {Neural Computation}, +volume = {20}, +number = {6}, +pages = {1631-1649}, +year = {2008}, +doi = {10.1162/neco.2008.04-07-510}, + +URL = { + https://doi.org/10.1162/neco.2008.04-07-510 + +}, +eprint = { + https://doi.org/10.1162/neco.2008.04-07-510 + +} +, + abstract = { Deep belief networks (DBN) are generative neural network models with many layers of hidden explanatory factors, recently introduced by Hinton, Osindero, and Teh (2006) along with a greedy layer-wise unsupervised learning algorithm. The building block of a DBN is a probabilistic model called a restricted Boltzmann machine (RBM), used to represent one layer of the model. Restricted Boltzmann machines are interesting because inference is easy in them and because they have been successfully used as building blocks for training deeper models. We first prove that adding hidden units yields strictly improved modeling power, while a second theorem shows that RBMs are universal approximators of discrete distributions. We then study the question of whether DBNs with more layers are strictly more powerful in terms of representational power. This suggests a new and less greedy criterion for training RBMs within DBNs. } +} + + + +@article{PhysRevLett.120.190501, + title = {Fidelity Witnesses for Fermionic Quantum Simulations}, + author = {Gluza, M. and Kliesch, M. and Eisert, J. and Aolita, L.}, + journal = {Phys. Rev. Lett.}, + volume = {120}, + issue = {19}, + pages = {190501}, + numpages = {7}, + year = {2018}, + month = {May}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.120.190501}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.120.190501} +} + +@article{aolita_reliable_2015, + title = {Reliable quantum certification of photonic state preparations}, + volume = {6}, + issn = {2041-1723}, + url = {https://doi.org/10.1038/ncomms9498}, + doi = {10.1038/ncomms9498}, + abstract = {Quantum technologies promise a variety of exciting applications. Even though impressive progress has been achieved recently, a major bottleneck currently is the lack of practical certification techniques. The challenge consists of ensuring that classically intractable quantum devices perform as expected. Here we present an experimentally friendly and reliable certification tool for photonic quantum technologies: an efficient certification test for experimental preparations of multimode pure Gaussian states, pure non-Gaussian states generated by linear-optical circuits with Fock-basis states of constant boson number as inputs, and pure states generated from the latter class by post-selecting with Fock-basis measurements on ancillary modes. Only classical computing capabilities and homodyne or hetorodyne detection are required. Minimal assumptions are made on the noise or experimental capabilities of the preparation. The method constitutes a step forward in many-body quantum certification, which is ultimately about testing quantum mechanics at large scales.}, + number = {1}, + journal = {Nature Communications}, + author = {Aolita, Leandro and Gogolin, Christian and Kliesch, Martin and Eisert, Jens}, + month = nov, + year = {2015}, + pages = {8498}, +} + +@article{PhysRevLett.106.230501, + title = {Direct Fidelity Estimation from Few Pauli Measurements}, + author = {Flammia, Steven T. and Liu, Yi-Kai}, + journal = {Phys. Rev. Lett.}, + volume = {106}, + issue = {23}, + pages = {230501}, + numpages = {4}, + year = {2011}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.106.230501}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.106.230501} +} + + + +@article{PhysRevA.99.052350, + title = {Statistical analysis of randomized benchmarking}, + author = {Harper, Robin and Hincks, Ian and Ferrie, Chris and Flammia, Steven T. and Wallman, Joel J.}, + journal = {Phys. Rev. A}, + volume = {99}, + issue = {5}, + pages = {052350}, + numpages = {7}, + year = {2019}, + month = {May}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.99.052350}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.99.052350} +} + + + +@article{PhysRevA.85.042311, + title = {Characterizing quantum gates via randomized benchmarking}, + author = {Magesan, Easwar and Gambetta, Jay M. and Emerson, Joseph}, + journal = {Phys. Rev. A}, + volume = {85}, + issue = {4}, + pages = {042311}, + numpages = {16}, + year = {2012}, + month = {Apr}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.85.042311}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.85.042311} +} + + +@article{PhysRevA.77.012307, + title = {Randomized benchmarking of quantum gates}, + author = {Knill, E. and Leibfried, D. and Reichle, R. and Britton, J. and Blakestad, R. B. and Jost, J. D. and Langer, C. and Ozeri, R. and Seidelin, S. and Wineland, D. J.}, + journal = {Phys. Rev. A}, + volume = {77}, + issue = {1}, + pages = {012307}, + numpages = {7}, + year = {2008}, + month = {Jan}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.77.012307}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.77.012307} +} + + +@article{PhysRevLett.105.250403, + title = {Permutationally Invariant Quantum Tomography}, + author = {T\'oth, G. and Wieczorek, W. and Gross, D. and Krischek, R. and Schwemmer, C. and Weinfurter, H.}, + journal = {Phys. Rev. Lett.}, + volume = {105}, + issue = {25}, + pages = {250403}, + numpages = {4}, + year = {2010}, + month = {Dec}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.105.250403}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.105.250403} +} + + + +@article{Moroder_2012, + doi = {10.1088/1367-2630/14/10/105001}, + url = {https://doi.org/10.1088%2F1367-2630%2F14%2F10%2F105001}, + year = 2012, + month = {oct}, + publisher = {{IOP} Publishing}, + volume = {14}, + number = {10}, + pages = {105001}, + author = {Tobias Moroder and Philipp Hyllus and G{\'{e}}za T{\'{o}}th and Christian Schwemmer and Alexander Niggebaum and Stefanie Gaile and Otfried Gühne and Harald Weinfurter}, + title = {Permutationally invariant state reconstruction}, + journal = {New Journal of Physics}, + abstract = {Feasible tomography schemes for large particle numbers must possess, besides an appropriate data acquisition protocol, an efficient way to reconstruct the density operator from the observed finite data set. Since state reconstruction typically requires the solution of a nonlinear large-scale optimization problem, this is a major challenge in the design of scalable tomography schemes. Here we present an efficient state reconstruction scheme for permutationally invariant quantum state tomography. It works for all common state-of-the-art reconstruction principles, including, in particular, maximum likelihood and least squares methods, which are the preferred choices in today's experiments. This high efficiency is achieved by greatly reducing the dimensionality of the problem employing a particular representation of permutationally invariant states known from spin coupling combined with convex optimization, which has clear advantages regarding speed, control and accuracy in comparison to commonly employed numerical routines. First prototype implementations easily allow reconstruction of a state of 20 qubits in a few minutes on a standard computer.} +} + +@article{doi:10.1146/annurev-conmatphys-031218-013401, +author = {Knolle, J. and Moessner, R.}, +title = {A Field Guide to Spin Liquids}, +journal = {Annual Review of Condensed Matter Physics}, +volume = {10}, +number = {1}, +pages = {451-472}, +year = {2019}, +doi = {10.1146/annurev-conmatphys-031218-013401}, + +URL = { + https://doi.org/10.1146/annurev-conmatphys-031218-013401 + +}, +eprint = { + https://doi.org/10.1146/annurev-conmatphys-031218-013401 + +} +, + abstract = { Spin liquids are collective phases of quantum matter that have eluded discovery in correlated magnetic materials for over half a century. Theoretical models of these enigmatic topological phases are no longer in short supply. In experiment there also exist plenty of promising candidate materials for their realization. One of the central challenges for the clear diagnosis of a spin liquid has been to connect the two. From that perspective, this review discusses characteristic features in experiment, resulting from the unusual properties of spin liquids. This takes us to thermodynamic, spectroscopic, transport, and other experiments on a search for traces of emergent gauge fields, spinons, Majorana fermions, and other fractionalized particles. } +} + + + + +@article{savaryQuantumSpinLiquids2016, + title = {Quantum Spin Liquids: A Review}, + shorttitle = {Quantum Spin Liquids}, + author = {Savary, Lucile and Balents, Leon}, + year = {2016}, + month = nov, + volume = {80}, + pages = {016502}, + issn = {0034-4885}, + doi = {10.1088/0034-4885/80/1/016502}, + abstract = {Quantum spin liquids may be considered `quantum disordered' ground states of spin systems, in which zero-point fluctuations are so strong that they prevent conventional magnetic long-range order. More interestingly, quantum spin liquids are prototypical examples of ground states with massive many-body entanglement, which is of a degree sufficient to render these states distinct phases of matter. Their highly entangled nature imbues quantum spin liquids with unique physical aspects, such as non-local excitations, topological properties, and more. In this review, we discuss the nature of such phases and their properties based on paradigmatic models and general arguments, and introduce theoretical technology such as gauge theory and partons, which are conveniently used in the study of quantum spin liquids. An overview is given of the different types of quantum spin liquids and the models and theories used to describe them. We also provide a guide to the current status of experiments in relation to study quantum spin liquids, and to the diverse probes used therein.}, + file = {/Users/carrasqu/Zotero/storage/3BN7MBES/Savary and Balents - 2016 - Quantum spin liquids a review.pdf}, + journal = {Reports on Progress in Physics}, + number = {1} +} + + + + +@article{RevModPhys.91.021001, + title = {Colloquium: Many-body localization, thermalization, and entanglement}, + author = {Abanin, Dmitry A. and Altman, Ehud and Bloch, Immanuel and Serbyn, Maksym}, + journal = {Rev. Mod. Phys.}, + volume = {91}, + issue = {2}, + pages = {021001}, + numpages = {26}, + year = {2019}, + month = {May}, + publisher = {American Physical Society}, + doi = {10.1103/RevModPhys.91.021001}, + url = {https://link.aps.org/doi/10.1103/RevModPhys.91.021001} +} + + +@article{basko2006, + title = {Metal\textendash Insulator Transition in a Weakly Interacting Many-Electron System with Localized Single-Particle States}, + author = {Basko, D. M. and Aleiner, I. L. and Altshuler, B. L.}, + year = {2006}, + month = may, + volume = {321}, + pages = {1126--1205}, + issn = {0003-4916}, + doi = {10.1016/j.aop.2005.11.014}, + abstract = {We consider low-temperature behavior of weakly interacting electrons in disordered conductors in the regime when all single-particle eigenstates are localized by the quenched disorder. We prove that in the absence of coupling of the electrons to any external bath dc electrical conductivity exactly vanishes as long as the temperature T does not exceed some finite value Tc. At the same time, it can be also proven that at high enough T the conductivity is finite. These two statements imply that the system undergoes a finite temperature metal-to-insulator transition, which can be viewed as Anderson-like localization of many-body wave functions in the Fock space. Metallic and insulating states are not different from each other by any spatial or discrete symmetries. We formulate the effective Hamiltonian description of the system at low energies (of the order of the level spacing in the single-particle localization volume). In the metallic phase quantum Boltzmann equation is valid, allowing to find the kinetic coefficients. In the insulating phase, T}, + file = {/Users/carrasqu/Zotero/storage/GUKFC79I/Basko et al. - 2006 - Metal–insulator transition in a weakly interacting.pdf}, + journal = {Annals of Physics}, + keywords = {Anderson localization,Fock space,Metal–insulator transition}, + number = {5} +} + + + + +@book{Goodfellow-et-al-2016, + title={Deep Learning}, + author={Ian Goodfellow and Yoshua Bengio and Aaron Courville}, + publisher={MIT Press}, + note={\url{http://www.deeplearningbook.org}}, + year={2016} +} + +@book{Xiao:803748, + author = "Wen Xiao Gang", + title = "{Quantum field theory of many-body systems: from the + origin of sound to an origin of light and electrons}", + publisher = "Oxford University Press", + address = "Oxford", + year = "2007", + url = "https://cds.cern.ch/record/803748", + doi = "10.1093/acprof:oso/9780199227259.001.0001", +} + +@article{inack2018, + title = {Projective quantum Monte Carlo simulations guided by unrestricted neural network states}, + author = {Inack, E. M. and Santoro, G. E. and Dell'Anna, L. and Pilati, S.}, + journal = {Phys. Rev. B}, + volume = {98}, + issue = {23}, + pages = {235145}, + numpages = {9}, + year = {2018}, + month = {Dec}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.98.235145}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.98.235145} +} + +@article{huang2017, + title = {Accelerated Monte Carlo simulations with restricted Boltzmann machines}, + author = {Huang, Li and Wang, Lei}, + journal = {Phys. Rev. B}, + volume = {95}, + issue = {3}, + pages = {035105}, + numpages = {6}, + year = {2017}, + month = {Jan}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.95.035105}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.95.035105} +} + +@article{mcnaughton2020, + title = {Boosting Monte Carlo simulations of spin glasses using autoregressive neural networks}, + author = {McNaughton, B. and Milo\ifmmode \check{s}\else \v{s}\fi{}evi\ifmmode \acute{c}\else \'{c}\fi{}, M. V. and Perali, A. and Pilati, S.}, + journal = {Phys. Rev. E}, + volume = {101}, + issue = {5}, + pages = {053312}, + numpages = {12}, + year = {2020}, + month = {May}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevE.101.053312}, + url = {https://link.aps.org/doi/10.1103/PhysRevE.101.053312} +} + + +@article{xiao_yan2017, + title = {Self-learning quantum Monte Carlo method in interacting fermion systems}, + author = {Xu, Xiao Yan and Qi, Yang and Liu, Junwei and Fu, Liang and Meng, Zi Yang}, + journal = {Phys. Rev. B}, + volume = {96}, + issue = {4}, + pages = {041119}, + numpages = {5}, + year = {2017}, + month = {Jul}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.96.041119}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.96.041119} +} +@article{junwei2017, + title = {Self-learning Monte Carlo method}, + author = {Liu, Junwei and Qi, Yang and Meng, Zi Yang and Fu, Liang}, + journal = {Phys. Rev. B}, + volume = {95}, + issue = {4}, + pages = {041101}, + numpages = {5}, + year = {2017}, + month = {Jan}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.95.041101}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.95.041101} +} + + +@article{parolini2019, + title = {Tunneling in projective quantum Monte Carlo simulations with guiding wave functions}, + author = {Parolini, T. and Inack, E. M. and Giudici, G. and Pilati, S.}, + journal = {Phys. Rev. B}, + volume = {100}, + issue = {21}, + pages = {214303}, + numpages = {10}, + year = {2019}, + month = {Dec}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.100.214303}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.100.214303} +} + +@article{Wu_2019, + title = {Solving Statistical Mechanics Using Variational Autoregressive Networks}, + author = {Wu, Dian and Wang, Lei and Zhang, Pan}, + journal = {Phys. Rev. Lett.}, + volume = {122}, + issue = {8}, + pages = {080602}, + numpages = {6}, + year = {2019}, + month = {Feb}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.122.080602}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.122.080602} +} + + +@article{albergo2019, + title = {Flow-based generative models for Markov chain Monte Carlo in lattice field theory}, + author = {Albergo, M. S. and Kanwar, G. and Shanahan, P. E.}, + journal = {Phys. Rev. D}, + volume = {100}, + issue = {3}, + pages = {034515}, + numpages = {13}, + year = {2019}, + month = {Aug}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevD.100.034515}, + url = {https://link.aps.org/doi/10.1103/PhysRevD.100.034515} +} + + +@article{pilati2019, + title = {Self-learning projective quantum Monte Carlo simulations guided by restricted Boltzmann machines}, + author = {Pilati, S. and Inack, E. M. and Pieri, P.}, + journal = {Phys. Rev. E}, + volume = {100}, + issue = {4}, + pages = {043301}, + numpages = {12}, + year = {2019}, + month = {Oct}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevE.100.043301}, + url = {https://link.aps.org/doi/10.1103/PhysRevE.100.043301} +} + + +@article{PhysRevLett.122.250502, + title = {Neural-Network Approach to Dissipative Quantum Many-Body Dynamics}, + author = {Hartmann, Michael J. and Carleo, Giuseppe}, + journal = {Phys. Rev. Lett.}, + volume = {122}, + issue = {25}, + pages = {250502}, + numpages = {6}, + year = {2019}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.122.250502}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.122.250502} +} + +@article{hermann2019deep, + title = {Deep-neural-network solution of the electronic {Schrödinger} equation}, + volume = {12}, + issn = {1755-4349}, + url = {https://doi.org/10.1038/s41557-020-0544-y}, + doi = {10.1038/s41557-020-0544-y}, + abstract = {The electronic Schrödinger equation can only be solved analytically for the hydrogen atom, and the numerically exact full configuration-interaction method is exponentially expensive in the number of electrons. Quantum Monte Carlo methods are a possible way out: they scale well for large molecules, they can be parallelized and their accuracy has, as yet, been only limited by the flexibility of the wavefunction ansatz used. Here we propose PauliNet, a deep-learning wavefunction ansatz that achieves nearly exact solutions of the electronic Schrödinger equation for molecules with up to 30 electrons. PauliNet has a multireference Hartree–Fock solution built in as a baseline, incorporates the physics of valid wavefunctions and is trained using variational quantum Monte Carlo. PauliNet outperforms previous state-of-the-art variational ansatzes for atoms, diatomic molecules and a strongly correlated linear H10, and matches the accuracy of highly specialized quantum chemistry methods on the transition-state energy of cyclobutadiene, while being computationally efficient.}, + number = {10}, + journal = {Nature Chemistry}, + author = {Hermann, Jan and Schätzle, Zeno and Noé, Frank}, + month = oct, + year = {2020}, + pages = {891--897}, +} + +@article{zi2018, + title = {Approximating quantum many-body wave functions using artificial neural networks}, + author = {Cai, Zi and Liu, Jinguo}, + journal = {Phys. Rev. B}, + volume = {97}, + issue = {3}, + pages = {035116}, + numpages = {8}, + year = {2018}, + month = {Jan}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.97.035116}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.97.035116} +} + +@article{AD_2017, +author = {Baydin, Atilim Gunes and Pearlmutter, Barak A. and Radul, Alexey Andreyevich and Siskind, Jeffrey Mark}, +title = {Automatic Differentiation in Machine Learning: A Survey}, +year = {2017}, +issue_date = {January 2017}, +volume = {18}, +number = {1}, +issn = {1532-4435}, +journal = {J. Mach. Learn. Res.} +} + + + +@article{Zhang_MLcuprates, + Abstract = {For centuries, the scientific discovery process has been based on systematic human observation and analysis of natural phenomena1. Today, however, automated instrumentation and large-scale data acquisition are generating datasets of such large volume and complexity as to defy conventional scientific methodology. Radically different scientific approaches are needed, and machine learning (ML) shows great promise for research fields such as materials science2--5. Given the success of ML in the analysis of synthetic data representing electronic quantum matter (EQM)6--16, the next challenge is to apply this approach to experimental data---for example, to the arrays of complex electronic-structure images17 obtained from atomic-scale visualization of EQM. Here we report the development and training of a suite of artificial neural networks (ANNs) designed to recognize different types of order hidden in such EQM image arrays. These ANNs are used to analyse an archive of experimentally derived EQM image arrays from carrier-doped copper oxide Mott insulators. In these noisy and complex data, the ANNs discover the existence of a lattice-commensurate, four-unit-cell periodic, translational-symmetry-breaking EQM state. Further, the ANNs determine that this state is unidirectional, revealing a coincident nematic EQM state. Strong-coupling theories of electronic liquid crystals18,19 are consistent with these observations.}, + Author = {Zhang, Yi and Mesaros, A. and Fujita, K. and Edkins, S. D. and Hamidian, M. H. and Ch'ng, K. and Eisaki, H. and Uchida, S. and Davis, J. C. S{\'e}amus and Khatami, Ehsan and Kim, Eun-Ah}, + Da = {2019/06/01}, + Date-Added = {2019-07-31 14:52:47 +0000}, + Date-Modified = {2019-07-31 14:52:47 +0000}, + Doi = {10.1038/s41586-019-1319-8}, + Id = {Zhang2019}, + Isbn = {1476-4687}, + Journal = {Nature}, + Number = {7762}, + Pages = {484--490}, + Title = {Machine learning in electronic-quantum-matter imaging experiments}, + Ty = {JOUR}, + Url = {https://doi.org/10.1038/s41586-019-1319-8}, + Volume = {570}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1038/s41586-019-1319-8}} + + +@article{leiwang2016, + title = {Discovering phase transitions with unsupervised learning}, + author = {Wang, Lei}, + journal = {Phys. Rev. B}, + volume = {94}, + issue = {19}, + pages = {195105}, + numpages = {5}, + year = {2016}, + month = {Nov}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.94.195105}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.94.195105} +} + +@article{Bohrdt2018, +abstract = {Quantum gas microscopes for ultracold atoms can provide high-resolution real-space snapshots of complex many-body systems. We implement machine learning to analyze and classify such snapshots of ultracold atoms. Specifically, we compare the data from an experimental realization of the two-dimensional Fermi-Hubbard model to two theoretical approaches: a doped quantum spin liquid state of resonating valence bond type, and the geometric string theory, describing a state with hidden spin order. This approach considers all available information without a potential bias towards one particular theory by the choice of an observable and can therefore select the theory which is more predictive in general. Up to intermediate doping values, our algorithm tends to classify experimental snapshots as geometric-string-like, as compared to the doped spin liquid. Our results demonstrate the potential for machine learning in processing the wealth of data obtained through quantum gas microscopy for new physical insights.}, +archivePrefix = {arXiv}, +arxivId = {1811.12425}, +author = {Bohrdt, Annabelle and Chiu, Christie S. and Ji, Geoffrey and Xu, Muqing and Greif, Daniel and Greiner, Markus and Demler, Eugene and Grusdt, Fabian and Knap, Michael}, +eprint = {1811.12425}, +file = {:Users/btimar/Library/Application Support/Mendeley Desktop/Downloaded/Bohrdt et al. - 2018 - Classifying Snapshots of the Doped Hubbard Model with Machine Learning.pdf:pdf}, +month = {nov}, +title = {{Classifying Snapshots of the Doped Hubbard Model with Machine Learning}}, +url = {http://arxiv.org/abs/1811.12425}, +year = {2018} +} + +@article{Rem2018, +abstract = {Machine learning techniques such as artificial neural networks are currently revolutionizing many technological areas and have also proven successful in quantum physics applications. Here we employ an artificial neural network and deep learning techniques to identify quantum phase transitions from single-shot experimental momentum-space density images of ultracold quantum gases and obtain results, which were not feasible with conventional methods. We map out the complete two-dimensional topological phase diagram of the Haldane model and provide an accurate characterization of the superfluid-to-Mott-insulator transition in an inhomogeneous Bose-Hubbard system. Our work points the way to unravel complex phase diagrams of general experimental systems, where the Hamiltonian and the order parameters might not be known.}, +archivePrefix = {arXiv}, +arxivId = {1809.05519}, +author = {Rem, Benno S. and K{\"{a}}ming, Niklas and Tarnowski, Matthias and Asteria, Luca and Fl{\"{a}}schner, Nick and Becker, Christoph and Sengstock, Klaus and Weitenberg, Christof}, +eprint = {1809.05519}, +file = {:Users/btimar/Library/Application Support/Mendeley Desktop/Downloaded/Rem et al. - 2018 - Identifying Quantum Phase Transitions using Artificial Neural Networks on Experimental Data.pdf:pdf}, +month = {sep}, +title = {{Identifying Quantum Phase Transitions using Artificial Neural Networks on Experimental Data}}, +url = {http://arxiv.org/abs/1809.05519}, +year = {2018} +} + +@article{evert2017nature, + Author = {van Nieuwenburg, Evert P. L. and Liu, Ye-Hua and Huber, Sebastian D.}, + Date = {2017/02/13/online}, + Date-Added = {2019-07-31 14:55:45 +0000}, + Date-Modified = {2019-07-31 14:55:45 +0000}, + Day = {13}, + Journal = {Nature Physics}, + L3 = {10.1038/nphys4037; }, + Month = {02}, + Pages = {435}, + Publisher = {Nature Publishing Group SN -}, + Title = {Learning phase transitions by confusion}, + Ty = {JOUR}, + Url = {https://doi.org/10.1038/nphys4037}, + Volume = {13}, + Year = {2017}, + Bdsk-Url-1 = {https://doi.org/10.1038/nphys4037}} + + +@ARTICLE{2020arXiv200905580L, + author = {{Luo}, Di and {Chen}, Zhuo and {Carrasquilla}, Juan and {Clark}, Bryan K.}, + title = "{Autoregressive Neural Network for Simulating Open Quantum Systems via a Probabilistic Formulation}", + journal = {arXiv e-prints}, + keywords = {Condensed Matter - Strongly Correlated Electrons, Condensed Matter - Disordered Systems and Neural Networks, Physics - Computational Physics, Quantum Physics}, + year = 2020, + month = sep, + eid = {arXiv:2009.05580}, + pages = {arXiv:2009.05580}, +archivePrefix = {arXiv}, + eprint = {2009.05580}, + primaryClass = {cond-mat.str-el}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv200905580L}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + + + +@ARTICLE{2019arXiv191211052C, + author = {{Carrasquilla}, Juan and {Luo}, Di and {P{\'e}rez}, Felipe and {Milsted}, Ashley and {Clark}, Bryan K. and {Volkovs}, Maksims and {Aolita}, Leandro}, + title = "{Probabilistic Simulation of Quantum Circuits with the Transformer}", + journal = {arXiv e-prints}, + keywords = {Condensed Matter - Strongly Correlated Electrons, Condensed Matter - Disordered Systems and Neural Networks, Quantum Physics}, + year = 2019, + month = dec, + eid = {arXiv:1912.11052}, + pages = {arXiv:1912.11052}, +archivePrefix = {arXiv}, + eprint = {1912.11052}, + primaryClass = {cond-mat.str-el}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2019arXiv191211052C}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + + +@article{kandala_hardware-efficient_2017, + title = {Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets}, + volume = {549}, + issn = {1476-4687}, + url = {https://doi.org/10.1038/nature23879}, + doi = {10.1038/nature23879}, + abstract = {The ground-state energy of small molecules is determined efficiently using six qubits of a superconducting quantum processor.}, + number = {7671}, + journal = {Nature}, + author = {Kandala, Abhinav and Mezzacapo, Antonio and Temme, Kristan and Takita, Maika and Brink, Markus and Chow, Jerry M. and Gambetta, Jay M.}, + month = sep, + year = {2017}, + pages = {242--246}, +} + + +@article{choo_fermionicnqs2020, + title = {Fermionic neural-network states for ab-initio electronic structure}, + volume = {11}, + issn = {2041-1723}, + url = {https://doi.org/10.1038/s41467-020-15724-9}, + doi = {10.1038/s41467-020-15724-9}, + abstract = {Neural-network quantum states have been successfully used to study a variety of lattice and continuous-space problems. Despite a great deal of general methodological developments, representing fermionic matter is however still early research activity. Here we present an extension of neural-network quantum states to model interacting fermionic problems. Borrowing techniques from quantum simulation, we directly map fermionic degrees of freedom to spin ones, and then use neural-network quantum states to perform electronic structure calculations. For several diatomic molecules in a minimal basis set, we benchmark our approach against widely used coupled cluster methods, as well as many-body variational states. On some test molecules, we systematically improve upon coupled cluster methods and Jastrow wave functions, reaching chemical accuracy or better. Finally, we discuss routes for future developments and improvements of the methods presented.}, + number = {1}, + journal = {Nature Communications}, + author = {Choo, Kenny and Mezzacapo, Antonio and Carleo, Giuseppe}, + month = may, + year = {2020}, + pages = {2368}, +} + + +@article{carrasquilla2017nature, + Author = {Carrasquilla, Juan and Melko, Roger G.}, + Date = {2017/02/13/online}, + Date-Added = {2019-07-31 14:55:21 +0000}, + Date-Modified = {2019-07-31 14:55:21 +0000}, + Day = {13}, + Journal = {Nature Physics}, + L3 = {10.1038/nphys4035; https://www.nature.com/articles/nphys4035#supplementary-information}, + Month = {02}, + Pages = {431}, + Publisher = {Nature Publishing Group SN -}, + Title = {Machine learning phases of matter}, + Ty = {JOUR}, + Url = {https://doi.org/10.1038/nphys4035}, + Volume = {13}, + Year = {2017}, + Bdsk-Url-1 = {https://doi.org/10.1038/nphys4035}} + +@article{androsiuk1993, +title = "Neural network solution of the Schroedinger equation for a two-dimensional harmonic oscillator", +journal = "Chemical Physics", +volume = "173", +number = "3", +pages = "377 - 383", +year = "1993", +issn = "0301-0104", +doi = "https://doi.org/10.1016/0301-0104(93)80153-Z", +url = "http://www.sciencedirect.com/science/article/pii/030101049380153Z", +author = "J. Androsiuk and L. Kulak and K. Sienicki", +} + + +@ARTICLE{LAGARIS19971, + + author={I. E. {Lagaris} and A. {Likas} and D. I. {Fotiadis}}, + + journal={IEEE Transactions on Neural Networks}, + + title={Artificial neural networks for solving ordinary and partial differential equations}, + + year={1998}, + + volume={9}, + + number={5}, + + pages={987-1000}, + + doi={10.1109/72.712178}} + + +@article {Carleo_2017, + author = {Carleo, Giuseppe and Troyer, Matthias}, + title = {Solving the quantum many-body problem with artificial neural networks}, + volume = {355}, + number = {6325}, + pages = {602--606}, + year = {2017}, + doi = {10.1126/science.aag2302}, + publisher = {American Association for the Advancement of Science}, + abstract = {Elucidating the behavior of quantum interacting systems of many particles remains one of the biggest challenges in physics. Traditional numerical methods often work well, but some of the most interesting problems leave them stumped. Carleo and Troyer harnessed the power of machine learning to develop a variational approach to the quantum many-body problem (see the Perspective by Hush). The method performed at least as well as state-of-the-art approaches, setting a benchmark for a prototypical two-dimensional problem. With further development, it may well prove a valuable piece in the quantum toolbox.Science, this issue p. 602; see also p. 580The challenge posed by the many-body problem in quantum physics originates from the difficulty of describing the nontrivial correlations encoded in the exponential complexity of the many-body wave function. Here we demonstrate that systematic machine learning of the wave function can reduce this complexity to a tractable computational form for some notable cases of physical interest. We introduce a variational representation of quantum states based on artificial neural networks with a variable number of hidden neurons. A reinforcement-learning scheme we demonstrate is capable of both finding the ground state and describing the unitary time evolution of complex interacting quantum systems. Our approach achieves high accuracy in describing prototypical interacting spins models in one and two dimensions.}, + issn = {0036-8075}, + journal = {Science} +} + + + +@article{PhysRevLett.122.250501, + title = {Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems}, + author = {Nagy, Alexandra and Savona, Vincenzo}, + journal = {Phys. Rev. Lett.}, + volume = {122}, + issue = {25}, + pages = {250501}, + numpages = {6}, + year = {2019}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.122.250501}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.122.250501} +} + +@article{Di_Luo, + title = {Backflow Transformations via Neural Networks for Quantum Many-Body Wave Functions}, + author = {Luo, Di and Clark, Bryan K.}, + journal = {Phys. Rev. Lett.}, + volume = {122}, + issue = {22}, + pages = {226401}, + numpages = {6}, + year = {2019}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.122.226401}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.122.226401} +} + +@article{pfau2019abinitio, + title = {Ab initio solution of the many-electron Schr\"odinger equation with deep neural networks}, + author = {Pfau, David and Spencer, James S. and Matthews, Alexander G. D. G. and Foulkes, W. M. C.}, + journal = {Phys. Rev. Research}, + volume = {2}, + issue = {3}, + pages = {033429}, + numpages = {20}, + year = {2020}, + month = {Sep}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevResearch.2.033429}, + url = {https://link.aps.org/doi/10.1103/PhysRevResearch.2.033429} +} + + + +@ARTICLE{roth2020iterative, + author = {{Roth}, Christopher}, + title = "{Iterative Retraining of Quantum Spin Models Using Recurrent Neural Networks}", + journal = {arXiv e-prints}, + keywords = {Physics - Computational Physics, Condensed Matter - Disordered Systems and Neural Networks, Condensed Matter - Strongly Correlated Electrons}, + year = 2020, + month = mar, + eid = {arXiv:2003.06228}, + pages = {arXiv:2003.06228}, +archivePrefix = {arXiv}, + eprint = {2003.06228}, + primaryClass = {physics.comp-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv200306228R}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + + +@article{RNNWF_2020, + title = {Recurrent neural network wave functions}, + author = {Hibat-Allah, Mohamed and Ganahl, Martin and Hayward, Lauren E. and Melko, Roger G. and Carrasquilla, Juan}, + journal = {Phys. Rev. Research}, + volume = {2}, + issue = {2}, + pages = {023358}, + numpages = {17}, + year = {2020}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevResearch.2.023358}, + url = {https://link.aps.org/doi/10.1103/PhysRevResearch.2.023358} +} + + +@article{PhysRevLett.124.020503, + title = {Deep Autoregressive Models for the Efficient Variational Simulation of Many-Body Quantum Systems}, + author = {Sharir, Or and Levine, Yoav and Wies, Noam and Carleo, Giuseppe and Shashua, Amnon}, + journal = {Phys. Rev. Lett.}, + volume = {124}, + issue = {2}, + pages = {020503}, + numpages = {6}, + year = {2020}, + month = {Jan}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.124.020503}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.124.020503} +} + + +@article{PhysRevLett.122.250503, + title = {Variational Neural-Network Ansatz for Steady States in Open Quantum Systems}, + author = {Vicentini, Filippo and Biella, Alberto and Regnault, Nicolas and Ciuti, Cristiano}, + journal = {Phys. Rev. Lett.}, + volume = {122}, + issue = {25}, + pages = {250503}, + numpages = {6}, + year = {2019}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.122.250503}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.122.250503} +} + + +@article{PhysRevB.99.214306, + title = {Constructing neural stationary states for open quantum many-body systems}, + author = {Yoshioka, Nobuyuki and Hamazaki, Ryusuke}, + journal = {Phys. Rev. B}, + volume = {99}, + issue = {21}, + pages = {214306}, + numpages = {8}, + year = {2019}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.99.214306}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.99.214306} +} + + + +@ARTICLE{2020arXiv201014510L, + author = {{Lode}, Axel U.~J. and {Lin}, Rui and {B{\"u}ttner}, Miriam and {Papariello}, Luca and {L{\'e}v{\^e}que}, Camille and {Chitra}, R. and {Tsatsos}, Marios C. and {Jaksch}, Dieter and {Molignini}, Paolo}, + title = "{Optimized Observable Readout from Single-shot Images of Ultracold Atoms via Machine Learning}", + journal = {arXiv e-prints}, + keywords = {Condensed Matter - Quantum Gases, Quantum Physics}, + year = 2020, + month = oct, + eid = {arXiv:2010.14510}, + pages = {arXiv:2010.14510}, +archivePrefix = {arXiv}, + eprint = {2010.14510}, + primaryClass = {cond-mat.quant-gas}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv201014510L}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + + +@article{PhysRevLett.122.210503, + title = {Machine Learning Topological Phases with a Solid-State Quantum Simulator}, + author = {Lian, Wenqian and Wang, Sheng-Tao and Lu, Sirui and Huang, Yuanyuan and Wang, Fei and Yuan, Xinxing and Zhang, Wengang and Ouyang, Xiaolong and Wang, Xin and Huang, Xianzhi and He, Li and Chang, Xiuying and Deng, Dong-Ling and Duan, Luming}, + journal = {Phys. Rev. Lett.}, + volume = {122}, + issue = {21}, + pages = {210503}, + numpages = {5}, + year = {2019}, + month = {May}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.122.210503}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.122.210503} +} + +@article{PhysRevB.99.121104, + title = {Machine learning of quantum phase transitions}, + author = {Dong, Xiao-Yu and Pollmann, Frank and Zhang, Xue-Feng}, + journal = {Phys. Rev. B}, + volume = {99}, + issue = {12}, + pages = {121104}, + numpages = {6}, + year = {2019}, + month = {Mar}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.99.121104}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.99.121104} +} + +@article{berezutskii2020, + title = {Probing Criticality in Quantum Spin Chains with Neural Networks}, + author = {Berezutskii, A. and Beketov, M. and Yudin, D. and Zimbor{\'a}s, Z. and Biamonte, J. D.}, + year = {2020}, + month = aug, + volume = {1}, + pages = {03LT01}, + publisher = {{IOP Publishing}}, + issn = {2632-072X}, + doi = {10.1088/2632-072X/abaa2b}, + abstract = {The numerical emulation of quantum systems often requires an exponential number of degrees of freedom which translates to a computational bottleneck. Methods of machine learning have been used in adjacent fields for effective feature extraction and dimensionality reduction of high-dimensional datasets. Recent studies have revealed that neural networks are further suitable for the determination of macroscopic phases of matter and associated phase transitions as well as efficient quantum state representation. In this work, we address quantum phase transitions in quantum spin chains, namely the transverse field Ising chain and the anisotropic XY chain, and show that even neural networks with no hidden layers can be effectively trained to distinguish between magnetically ordered and disordered phases. Our neural network acts to predict the corresponding crossovers finite-size systems undergo. Our results extend to a wide class of interacting quantum many-body systems and illustrate the wide applicability of neural networks to many-body quantum physics.}, + journal = {Journal of Physics: Complexity}, + number = {3} +} + + + +@article{PhysRevLett.125.170603, + title = {Unsupervised Phase Discovery with Deep Anomaly Detection}, + author = {Kottmann, Korbinian and Huembeli, Patrick and Lewenstein, Maciej and Ac\'{\i}n, Antonio}, + journal = {Phys. Rev. Lett.}, + volume = {125}, + issue = {17}, + pages = {170603}, + numpages = {6}, + year = {2020}, + month = {Oct}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.125.170603}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.125.170603} +} + +@article{PhysRevB.102.054512, + title = {Deep learning of topological phase transitions from entanglement aspects}, + author = {Tsai, Yuan-Hong and Yu, Meng-Zhe and Hsu, Yu-Hao and Chung, Ming-Chiang}, + journal = {Phys. Rev. B}, + volume = {102}, + issue = {5}, + pages = {054512}, + numpages = {8}, + year = {2020}, + month = {Aug}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.102.054512}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.102.054512} +} + +@article{PhysRevB.99.060404, + title = {Probing hidden spin order with interpretable machine learning}, + author = {Greitemann, Jonas and Liu, Ke and Pollet, Lode}, + journal = {Phys. Rev. B}, + volume = {99}, + issue = {6}, + pages = {060404}, + numpages = {6}, + year = {2019}, + month = {Feb}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.99.060404}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.99.060404} +} + +@article{PhysRevB.99.104410, + title = {Learning multiple order parameters with interpretable machines}, + author = {Liu, Ke and Greitemann, Jonas and Pollet, Lode}, + journal = {Phys. Rev. B}, + volume = {99}, + issue = {10}, + pages = {104410}, + numpages = {15}, + year = {2019}, + month = {Mar}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.99.104410}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.99.104410} +} + +@article{PhysRevLett.124.010508, + title = {Discovering Physical Concepts with Neural Networks}, + author = {Iten, Raban and Metger, Tony and Wilming, Henrik and del Rio, L\'{\i}dia and Renner, Renato}, + journal = {Phys. Rev. Lett.}, + volume = {124}, + issue = {1}, + pages = {010508}, + numpages = {6}, + year = {2020}, + month = {Jan}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.124.010508}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.124.010508} +} + + +@article{PhysRevX.10.011006, + title = {Using a Recurrent Neural Network to Reconstruct Quantum Dynamics of a Superconducting Qubit from Physical Observations}, + author = {Flurin, E. and Martin, L. S. and Hacohen-Gourgy, S. and Siddiqi, I.}, + journal = {Phys. Rev. X}, + volume = {10}, + issue = {1}, + pages = {011006}, + numpages = {10}, + year = {2020}, + month = {Jan}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevX.10.011006}, + url = {https://link.aps.org/doi/10.1103/PhysRevX.10.011006} +} + + + +@article{PhysRevE.99.062106, + title = {Generation of ice states through deep reinforcement learning}, + author = {Zhao, Kai-Wen and Kao, Wen-Han and Wu, Kai-Hsin and Kao, Ying-Jer}, + journal = {Phys. Rev. E}, + volume = {99}, + issue = {6}, + pages = {062106}, + numpages = {9}, + year = {2019}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevE.99.062106}, + url = {https://link.aps.org/doi/10.1103/PhysRevE.99.062106} +} + +@ARTICLE{2014arXiv1410.3831M, + author = {{Mehta}, Pankaj and {Schwab}, David J.}, + title = "{An exact mapping between the Variational Renormalization Group and Deep Learning}", + journal = {arXiv e-prints}, + keywords = {Statistics - Machine Learning, Condensed Matter - Statistical Mechanics, Computer Science - Machine Learning, Computer Science - Neural and Evolutionary Computing}, + year = 2014, + month = oct, + eid = {arXiv:1410.3831}, + pages = {arXiv:1410.3831}, +archivePrefix = {arXiv}, + eprint = {1410.3831}, + primaryClass = {stat.ML}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2014arXiv1410.3831M}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@article{PhysRevB.97.045153, + title = {Machine learning spatial geometry from entanglement features}, + author = {You, Yi-Zhuang and Yang, Zhao and Qi, Xiao-Liang}, + journal = {Phys. Rev. B}, + volume = {97}, + issue = {4}, + pages = {045153}, + numpages = {13}, + year = {2018}, + month = {Jan}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.97.045153}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.97.045153} +} + + +@article{koch-janusz_mutual_2018, + title = {Mutual information, neural networks and the renormalization group}, + volume = {14}, + issn = {1745-2481}, + url = {https://doi.org/10.1038/s41567-018-0081-4}, + doi = {10.1038/s41567-018-0081-4}, + abstract = {Physical systems differing in their microscopic details often display strikingly similar behaviour when probed at macroscopic scales. Those universal properties, largely determining their physical characteristics, are revealed by the powerful renormalization group (RG) procedure, which systematically retains ‘slow’ degrees of freedom and integrates out the rest. However, the important degrees of freedom may be difficult to identify. Here we demonstrate a machine-learning algorithm capable of identifying the relevant degrees of freedom and executing RG steps iteratively without any prior knowledge about the system. We introduce an artificial neural network based on a model-independent, information-theoretic characterization of a real-space RG procedure, which performs this task. We apply the algorithm to classical statistical physics problems in one and two dimensions. We demonstrate RG flow and extract the Ising critical exponent. Our results demonstrate that machine-learning techniques can extract abstract physical concepts and consequently become an integral part of theory- and model-building.}, + number = {6}, + journal = {Nature Physics}, + author = {Koch-Janusz, Maciej and Ringel, Zohar}, + month = jun, + year = {2018}, + pages = {578--582}, +} + +@article{PhysRevE.97.053304, + title = {Scale-invariant feature extraction of neural network and renormalization group flow}, + author = {Iso, Satoshi and Shiba, Shotaro and Yokoo, Sumito}, + journal = {Phys. Rev. E}, + volume = {97}, + issue = {5}, + pages = {053304}, + numpages = {16}, + year = {2018}, + month = {May}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevE.97.053304}, + url = {https://link.aps.org/doi/10.1103/PhysRevE.97.053304} +} + +@article{PhysRevLett.121.260601, + title = {Neural Network Renormalization Group}, + author = {Li, Shuo-Hui and Wang, Lei}, + journal = {Phys. Rev. Lett.}, + volume = {121}, + issue = {26}, + pages = {260601}, + numpages = {7}, + year = {2018}, + month = {Dec}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.121.260601}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.121.260601} +} + + +@article{PhysRevResearch.2.023369, + title = {Machine learning holographic mapping by neural network renormalization group}, + author = {Hu, Hong-Ye and Li, Shuo-Hui and Wang, Lei and You, Yi-Zhuang}, + journal = {Phys. Rev. Research}, + volume = {2}, + issue = {2}, + pages = {023369}, + numpages = {12}, + year = {2020}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevResearch.2.023369}, + url = {https://link.aps.org/doi/10.1103/PhysRevResearch.2.023369} +} + +@ARTICLE{2020arXiv200600712H, + author = {{Hashimoto}, Koji and {Hu}, Hong-Ye and {You}, Yi-Zhuang}, + title = "{Neural ODE and Holographic QCD}", + journal = {arXiv e-prints}, + keywords = {High Energy Physics - Theory, Condensed Matter - Disordered Systems and Neural Networks, General Relativity and Quantum Cosmology, High Energy Physics - Phenomenology}, + year = 2020, + month = jun, + eid = {arXiv:2006.00712}, + pages = {arXiv:2006.00712}, +archivePrefix = {arXiv}, + eprint = {2006.00712}, + primaryClass = {hep-th}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv200600712H}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@ARTICLE{2020arXiv201005703C, + author = {{Chung}, Jui-Hui and {Kao}, Ying-Jer}, + title = "{Neural Monte Carlo Renormalization Group}", + journal = {arXiv e-prints}, + keywords = {Condensed Matter - Disordered Systems and Neural Networks, Condensed Matter - Statistical Mechanics, High Energy Physics - Theory}, + year = 2020, + month = oct, + eid = {arXiv:2010.05703}, + pages = {arXiv:2010.05703}, +archivePrefix = {arXiv}, + eprint = {2010.05703}, + primaryClass = {cond-mat.dis-nn}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv201005703C}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@article{PhysRevX.8.031086, + title = {Reinforcement Learning in Different Phases of Quantum Control}, + author = {Bukov, Marin and Day, Alexandre G. R. and Sels, Dries and Weinberg, Phillip and Polkovnikov, Anatoli and Mehta, Pankaj}, + journal = {Phys. Rev. X}, + volume = {8}, + issue = {3}, + pages = {031086}, + numpages = {15}, + year = {2018}, + month = {Sep}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevX.8.031086}, + url = {https://link.aps.org/doi/10.1103/PhysRevX.8.031086} +} + +@article{PhysRevX.8.031084, + title = {Reinforcement Learning with Neural Networks for Quantum Feedback}, + author = {F\"osel, Thomas and Tighineanu, Petru and Weiss, Talitha and Marquardt, Florian}, + journal = {Phys. Rev. X}, + volume = {8}, + issue = {3}, + pages = {031084}, + numpages = {15}, + year = {2018}, + month = {Sep}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevX.8.031084}, + url = {https://link.aps.org/doi/10.1103/PhysRevX.8.031084} +} + +@article{PhysRevLett.122.020601, + title = {Glassy Phase of Optimal Quantum Control}, + author = {Day, Alexandre G. R. and Bukov, Marin and Weinberg, Phillip and Mehta, Pankaj and Sels, Dries}, + journal = {Phys. Rev. Lett.}, + volume = {122}, + issue = {2}, + pages = {020601}, + numpages = {6}, + year = {2019}, + month = {Jan}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.122.020601}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.122.020601} +} + +@article{huembeli2018, + title = {Identifying quantum phase transitions with adversarial neural networks}, + author = {Huembeli, Patrick and Dauphin, Alexandre and Wittek, Peter}, + journal = {Phys. Rev. B}, + volume = {97}, + issue = {13}, + pages = {134109}, + numpages = {9}, + year = {2018}, + month = {Apr}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.97.134109}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.97.134109} +} + + +@article{PhysRevA.102.033326, + title = {Visualizing strange metallic correlations in the two-dimensional Fermi-Hubbard model with artificial intelligence}, + author = {Khatami, Ehsan and Guardado-Sanchez, Elmer and Spar, Benjamin M. and Carrasquilla, Juan Felipe and Bakr, Waseem S. and Scalettar, Richard T.}, + journal = {Phys. Rev. A}, + volume = {102}, + issue = {3}, + pages = {033326}, + numpages = {11}, + year = {2020}, + month = {Sep}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.102.033326}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.102.033326} +} + + +@article{chng2017, + title = {Machine Learning Phases of Strongly Correlated Fermions}, + author = {Ch'ng, Kelvin and Carrasquilla, Juan and Melko, Roger G. and Khatami, Ehsan}, + journal = {Phys. Rev. X}, + volume = {7}, + issue = {3}, + pages = {031038}, + numpages = {9}, + year = {2017}, + month = {Aug}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevX.7.031038}, + url = {https://link.aps.org/doi/10.1103/PhysRevX.7.031038} +} + + +@article{PhysRevLett.125.127401, + title = {Identifying Topological Phase Transitions in Experiments Using Manifold Learning}, + author = {Lustig, Eran and Yair, Or and Talmon, Ronen and Segev, Mordechai}, + journal = {Phys. Rev. Lett.}, + volume = {125}, + issue = {12}, + pages = {127401}, + numpages = {6}, + year = {2020}, + month = {Sep}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.125.127401}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.125.127401} +} + + + +@article{rodriguez-nieva2019, + title = {Identifying Topological Order through Unsupervised Machine Learning}, + author = {{Rodriguez-Nieva}, Joaquin F. and Scheurer, Mathias S.}, + year = {2019}, + month = aug, + volume = {15}, + pages = {790--795}, + issn = {1745-2481}, + doi = {10.1038/s41567-019-0512-x}, + abstract = {Machine learning techniques have latterly gained currency in condensed-matter physics, for example by identifying phase transitions. An unsupervised machine learning algorithm that identifies topological order is now demonstrated.}, + copyright = {2019 The Author(s), under exclusive licence to Springer Nature Limited}, + journal = {Nature Physics}, + number = {8} +} + + + + +@article{broecker2017b, + title = {Quantum Phase Recognition via Unsupervised Machine Learning}, + author = {Broecker, Peter and Assaad, Fakher F. and Trebst, Simon}, + year = {2017}, + month = jul, + abstract = {The application of state-of-the-art machine learning techniques to statistical physic problems has seen a surge of interest for their ability to discriminate phases of matter by extracting essential features in the many-body wavefunction or the ensemble of correlators sampled in Monte Carlo simulations. Here we introduce a gener- alization of supervised machine learning approaches that allows to accurately map out phase diagrams of inter- acting many-body systems without any prior knowledge, e.g. of their general topology or the number of distinct phases. To substantiate the versatility of this approach, which combines convolutional neural networks with quantum Monte Carlo sampling, we map out the phase diagrams of interacting boson and fermion models both at zero and finite temperatures and show that first-order, second-order, and Kosterlitz-Thouless phase transitions can all be identified. We explicitly demonstrate that our approach is capable of identifying the phase transition to non-trivial many-body phases such as superfluids or topologically ordered phases without supervision.}, + archivePrefix = {arXiv}, + eprint = {1707.00663}, + eprinttype = {arxiv}, + journal = {arXiv:1707.00663 [cond-mat]}, + keywords = {Condensed Matter - Disordered Systems and Neural Networks,Condensed Matter - Statistical Mechanics,Condensed Matter - Strongly Correlated Electrons}, + primaryClass = {cond-mat} +} + + + + +@article{broecker2017, + title = {Machine learning quantum phases of matter beyond the fermion sign problem}, + volume = {7}, + issn = {2045-2322}, + url = {https://doi.org/10.1038/s41598-017-09098-0}, + doi = {10.1038/s41598-017-09098-0}, + abstract = {State-of-the-art machine learning techniques promise to become a powerful tool in statistical mechanics via their capacity to distinguish different phases of matter in an automated way. Here we demonstrate that convolutional neural networks (CNN) can be optimized for quantum many-fermion systems such that they correctly identify and locate quantum phase transitions in such systems. Using auxiliary-field quantum Monte Carlo (QMC) simulations to sample the many-fermion system, we show that the Green’s function holds sufficient information to allow for the distinction of different fermionic phases via a CNN. We demonstrate that this QMC + machine learning approach works even for systems exhibiting a severe fermion sign problem where conventional approaches to extract information from the Green’s function, e.g. in the form of equal-time correlation functions, fail.}, + number = {1}, + journal = {Scientific Reports}, + author = {Broecker, Peter and Carrasquilla, Juan and Melko, Roger G. and Trebst, Simon}, + month = aug, + year = {2017}, + pages = {8823}, +} + + +@ARTICLE{2020arXiv201003655Y, + author = {{Yao}, Jiahao and {Lin}, Lin and {Bukov}, Marin}, + title = "{Reinforcement Learning for Many-Body Ground State Preparation based on Counter-Diabatic Driving}", + journal = {arXiv e-prints}, + keywords = {Quantum Physics, Condensed Matter - Quantum Gases, Computer Science - Machine Learning, Physics - Computational Physics}, + year = 2020, + month = oct, + eid = {arXiv:2010.03655}, + pages = {arXiv:2010.03655}, +archivePrefix = {arXiv}, + eprint = {2010.03655}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv201003655Y}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@article{eun-ah2017, + title = {Quantum Loop Topography for Machine Learning}, + author = {Zhang, Yi and Kim, Eun-Ah}, + journal = {Phys. Rev. Lett.}, + volume = {118}, + issue = {21}, + pages = {216401}, + numpages = {5}, + year = {2017}, + month = {May}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.118.216401}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.118.216401} +} + +@article{dassarma2017, + title = {Machine learning topological states}, + author = {Deng, Dong-Ling and Li, Xiaopeng and Das Sarma, S.}, + journal = {Phys. Rev. B}, + volume = {96}, + issue = {19}, + pages = {195145}, + numpages = {11}, + year = {2017}, + month = {Nov}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.96.195145}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.96.195145} +} + +@article{neupeurt2017, + title = {Probing many-body localization with neural networks}, + author = {Schindler, Frank and Regnault, Nicolas and Neupert, Titus}, + journal = {Phys. Rev. B}, + volume = {95}, + issue = {24}, + pages = {245134}, + numpages = {11}, + year = {2017}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.95.245134}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.95.245134} +} + +@article{yi-ting2018, + title = {Machine Learning Many-Body Localization: Search for the Elusive Nonergodic Metal}, + author = {Hsu, Yi-Ting and Li, Xiao and Deng, Dong-Ling and Das Sarma, S.}, + journal = {Phys. Rev. Lett.}, + volume = {121}, + issue = {24}, + pages = {245701}, + numpages = {6}, + year = {2018}, + month = {Dec}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.121.245701}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.121.245701} +} + + + +@misc{tensorflow, +title={{TensorFlow}: Large-Scale Machine Learning on Heterogeneous Systems}, +url={http://tensorflow.org/}, +note={Software available from tensorflow.org}, +author={ + Mart\'{\i}n~Abadi and + Ashish~Agarwal and + Paul~Barham and + Eugene~Brevdo and + Zhifeng~Chen and + Craig~Citro and + Greg~S.~Corrado and + Andy~Davis and + Jeffrey~Dean and + Matthieu~Devin and + Sanjay~Ghemawat and + Ian~Goodfellow and + Andrew~Harp and + Geoffrey~Irving and + Michael~Isard and + Yangqing Jia and + Rafal~Jozefowicz and + Lukasz~Kaiser and + Manjunath~Kudlur and + Josh~Levenberg and + Dan~Man\'{e} and + Rajat~Monga and + Sherry~Moore and + Derek~Murray and + Chris~Olah and + Mike~Schuster and + Jonathon~Shlens and + Benoit~Steiner and + Ilya~Sutskever and + Kunal~Talwar and + Paul~Tucker and + Vincent~Vanhoucke and + Vijay~Vasudevan and + Fernanda~Vi\'{e}gas and + Oriol~Vinyals and + Pete~Warden and + Martin~Wattenberg and + Martin~Wicke and + Yuan~Yu and + Xiaoqiang~Zheng}, + year={2015}, +} + +@ARTICLE{itensor, + author = {{Fishman}, Matthew and {White}, Steven R. and {Miles Stoudenmire}, E.}, + title = "{The ITensor Software Library for Tensor Network Calculations}", + journal = {arXiv e-prints}, + keywords = {Computer Science - Mathematical Software, Condensed Matter - Strongly Correlated Electrons, Physics - Computational Physics}, + year = 2020, + month = jul, + eid = {arXiv:2007.14822}, + pages = {arXiv:2007.14822}, +archivePrefix = {arXiv}, + eprint = {2007.14822}, + primaryClass = {cs.MS}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv200714822F}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@article{Endres2016, +abstract = {The realization of large-scale fully controllable quantum systems is an exciting frontier in modern physical science. We use atom-by-atom assembly to implement a platform for the deterministic preparation of regular one-dimensional arrays of individually controlled cold atoms. In our approach, a measurement and feedback procedure emiliminates the entropy associated with probabilistic trap occupation and results in defect-free arrays of over 50 atoms in less than 400 milliseconds. The technique is based on fast, real-time control of 100 optical tweezers, which we use to arrange atoms in desired geometric patterns and to maintain these configurations by replacing lost atoms with surplus atoms from a reservoir. This bottom-up approach may enable controlled engineering of scalable many-body systems for quantum information processing, quantum simulations, and precision measurements.}, +author = {Endres, Manuel and Bernien, Hannes and Keesling, Alexander and Levine, Harry and Anschuetz, Eric R and Krajenbrink, Alexandre and Senko, Crystal and Vuletic, Vladan and Greiner, Markus and Lukin, Mikhail D}, +doi = {10.1126/science.aah3752}, +issn = {0036-8075}, +journal = {Science}, +month = {nov}, +number = {6315}, +pages = {1024--1027}, +publisher = {American Association for the Advancement of Science}, +title = {{Atom-by-atom assembly of defect-free one-dimensional cold atom arrays}}, +url = {http://www.sciencemag.org/lookup/doi/10.1126/science.aah3752}, +volume = {354}, +year = {2016} +} + +@article{Keesling2018, +abstract = {Quantum phase transitions (QPTs) involve transformations between different states of matter that are driven by quantum fluctuations. These fluctuations play a dominant role in the quantum critical region surrounding the transition point, where the dynamics are governed by the universal properties associated with the QPT. The resulting quantum criticality has been explored by probing linear response for systems near thermal equilibrium. While time dependent phenomena associated with classical phase transitions have been studied in various scientific fields, understanding critical real-time dynamics in isolated, non-equilibrium quantum systems is of fundamental importance both for exploring novel approaches to quantum information processing and realizing exotic new phases of matter. Here, we use a Rydberg atom quantum simulator with programmable interactions to study the quantum critical dynamics associated with several distinct QPTs. By studying the growth of spatial correlations while crossing the QPT at variable speeds, we experimentally verify the quantum Kibble-Zurek mechanism (QKZM) for an Ising-type QPT, explore scaling universality, and observe corrections beyond simple QKZM predictions. This approach is subsequently used to investigate novel QPTs associated with chiral clock model providing new insights into exotic systems, and opening the door for precision studies of critical phenomena and applications to quantum optimization.}, +archivePrefix = {arXiv}, +arxivId = {1809.05540}, +author = {Keesling, Alexander and Omran, Ahmed and Levine, Harry and Bernien, Hannes and Pichler, Hannes and Choi, Soonwon and Samajdar, Rhine and Schwartz, Sylvain and Silvi, Pietro and Sachdev, Subir and Zoller, Peter and Endres, Manuel and Greiner, Markus and Vuletic, Vladan and Lukin, Mikhail D.}, +eprint = {1809.05540}, +keywords = {Keesling2018}, +mendeley-tags = {Keesling2018}, +month = {sep}, +title = {{Probing quantum critical dynamics on a programmable Rydberg simulator}}, +url = {http://arxiv.org/abs/1809.05540}, +year = {2018} +} + + +@article{keesling_quantum_2019, + title = {Quantum Kibble-Zurek mechanism and critical dynamics on a programmable Rydberg simulator}, + volume = {568}, + issn = {1476-4687}, + url = {https://doi.org/10.1038/s41586-019-1070-1}, + doi = {10.1038/s41586-019-1070-1}, + number = {7751}, + journal = {Nature}, + author = {Keesling, Alexander and Omran, Ahmed and Levine, Harry and Bernien, Hannes and Pichler, Hannes and Choi, Soonwon and Samajdar, Rhine and Schwartz, Sylvain and Silvi, Pietro and Sachdev, Subir and Zoller, Peter and Endres, Manuel and Greiner, Markus and Vuletic, Vladan and Lukin, Mikhail D.}, + month = apr, + year = {2019}, + pages = {207--211}, +} + + +@article{Bernien2017, +annote = {Article}, +author = {Bernien, Hannes and Schwartz, Sylvain and Keesling, Alexander and Levine, Harry and Omran, Ahmed and Pichler, Hannes and Choi, Soonwon and Zibrov, Alexander S and Endres, Manuel and Greiner, Markus and Vuleti{\'{c}}, Vladan and Lukin, Mikhail D}, +doi = {10.1038/nature24622}, +issn = {0028-0836}, +journal = {Nature}, +month = {nov}, +number = {7682}, +pages = {579--584}, +publisher = {Macmillan Publishers Limited, part of Springer Nature. All rights reserved. SN -}, +title = {{Probing many-body dynamics on a 51-atom quantum simulator}}, +url = {http://dx.doi.org/10.1038/nature24622 http://www.nature.com/doifinder/10.1038/nature24622}, +volume = {551}, +year = {2017} +} + + +@article{Labuhn, + Author = {Labuhn, Henning and Barredo, Daniel and Ravets, Sylvain and de L{\'e}s{\'e}leuc, Sylvain and Macr{\`\i}, Tommaso and Lahaye, Thierry and Browaeys, Antoine}, + Date = {2016/06/01/online}, + Date-Added = {2019-03-29 14:59:41 +0000}, + Date-Modified = {2019-03-29 14:59:41 +0000}, + Day = {01}, + Journal = {Nature}, + L3 = {10.1038/nature18274; }, + Month = {06}, + Pages = {667 EP -}, + Publisher = {Nature Publishing Group, a division of Macmillan Publishers Limited. All Rights Reserved. SN -}, + Title = {Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models}, + Ty = {JOUR}, + Url = {https://doi.org/10.1038/nature18274}, + Volume = {534}, + Year = {2016}, + Bdsk-Url-1 = {https://doi.org/10.1038/nature18274}} + +@article {Omran570, + author = {Omran, A. and Levine, H. and Keesling, A. and Semeghini, G. and Wang, T. T. and Ebadi, S. and Bernien, H. and Zibrov, A. S. and Pichler, H. and Choi, S. and Cui, J. and Rossignolo, M. and Rembold, P. and Montangero, S. and Calarco, T. and Endres, M. and Greiner, M. and Vuleti{\'c}, V. and Lukin, M. D.}, + title = {Generation and manipulation of Schr{\"o}dinger cat states in Rydberg atom arrays}, + volume = {365}, + number = {6453}, + pages = {570--574}, + year = {2019}, + doi = {10.1126/science.aax9743}, + publisher = {American Association for the Advancement of Science}, + abstract = {The success of quantum computing relies on the ability to entangle large-scale systems. Various platforms are being pursued, with architectures based on superconducting qubits and trapped atoms being the most advanced. By entangling up to 20 qubits, Omran et al. and Song et al.{\textemdash}working with Rydberg atom qubits and superconducting qubits, respectively{\textemdash}demonstrate how far these platforms have reached. The demonstrated controllable generation and detection of entanglement on such quantum systems is promising for the development of large-scale quantum processors.Science, this issue p. 570, p. 574Quantum entanglement involving coherent superpositions of macroscopically distinct states is among the most striking features of quantum theory, but its realization is challenging because such states are extremely fragile. Using a programmable quantum simulator based on neutral atom arrays with interactions mediated by Rydberg states, we demonstrate the creation of {\textquotedblleft}Schr{\"o}dinger cat{\textquotedblright} states of the Greenberger-Horne-Zeilinger (GHZ) type with up to 20 qubits. Our approach is based on engineering the energy spectrum and using optimal control of the many-body system. We further demonstrate entanglement manipulation by using GHZ states to distribute entanglement to distant sites in the array, establishing important ingredients for quantum information processing and quantum metrology.}, + issn = {0036-8075}, + URL = {https://science.sciencemag.org/content/365/6453/570}, + eprint = {https://science.sciencemag.org/content/365/6453/570.full.pdf}, + journal = {Science} +} + + +@inproceedings{cho-etal-2014-learning, + title = "Learning Phrase Representations using {RNN} Encoder{--}Decoder for Statistical Machine Translation", + author = {Cho, Kyunghyun and + van Merri{\"e}nboer, Bart and + Gulcehre, Caglar and + Bahdanau, Dzmitry and + Bougares, Fethi and + Schwenk, Holger and + Bengio, Yoshua}, + booktitle = "Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing ({EMNLP})", + month = oct, + year = "2014", + address = "Doha, Qatar", + publisher = "Association for Computational Linguistics", + url = "https://www.aclweb.org/anthology/D14-1179", + doi = "10.3115/v1/D14-1179", + pages = "1724--1734", +} + +@article{Levine2019, + title = {Parallel Implementation of High-Fidelity Multiqubit Gates with Neutral Atoms}, + author = {Levine, Harry and Keesling, Alexander and Semeghini, Giulia and Omran, Ahmed and Wang, Tout T. and Ebadi, Sepehr and Bernien, Hannes and Greiner, Markus and Vuleti\ifmmode \acute{c}\else \'{c}\fi{}, Vladan and Pichler, Hannes and Lukin, Mikhail D.}, + journal = {Phys. Rev. Lett.}, + volume = {123}, + issue = {17}, + pages = {170503}, + numpages = {6}, + year = {2019}, + month = {Oct}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.123.170503}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.123.170503} +} + + + + +@article {Schauss1455, + author = {Schau{\ss}, P. and Zeiher, J. and Fukuhara, T. and Hild, S. and Cheneau, M. and Macr{\`\i}, T. and Pohl, T. and Bloch, I. and Gross, C.}, + title = {Crystallization in Ising quantum magnets}, + volume = {347}, + number = {6229}, + pages = {1455--1458}, + year = {2015}, + doi = {10.1126/science.1258351}, + publisher = {American Association for the Advancement of Science}, + abstract = {In physics, interactions between components of a system can cause it to become more orderly in an attempt to minimize energy. Such ordered phases appear, for example, in magnetic systems. Schauss et al. simulated these phenomena using a collection of neutral atoms at low temperatures. By shining laser light on the atoms, the authors brought some of them into a high energy state called the Rydberg state. By carefully varying the experimental parameters, they coaxed these Rydberg atoms into patterns reminiscent of crystal lattices in rod- and disk-shaped atomic samples.Science, this issue p. 1455 Dominating finite-range interactions in many-body systems can lead to intriguing self-ordered phases of matter. For quantum magnets, Ising models with power-law interactions are among the most elementary systems that support such phases. These models can be implemented by laser coupling ensembles of ultracold atoms to Rydberg states. Here, we report on the experimental preparation of crystalline ground states of such spin systems. We observe a magnetization staircase as a function of the system size and show directly the emergence of crystalline states with vanishing susceptibility. Our results demonstrate the precise control of Rydberg many-body systems and may enable future studies of phase transitions and quantum correlations in interacting quantum magnets.}, + issn = {0036-8075}, + journal = {Science} +} + + + +@article{Kullback:1951aa, + Author = {Kullback, S. and Leibler, R. A.}, + Da = {1951/03}, + Date-Added = {2020-05-06 19:36:51 +0000}, + Date-Modified = {2020-05-06 19:36:51 +0000}, + Doi = {10.1214/aoms/1177729694}, + Isbn = {0003-4851}, + Journal = {Ann. Math. Statist.}, + La = {en}, + Number = {1}, + Pages = {79--86}, + Publisher = {The Institute of Mathematical Statistics}, + Title = {On Information and Sufficiency}, + Ty = {JOUR}, + Url = {https://projecteuclid.org:443/euclid.aoms/1177729694}, + Volume = {22}, + Year = {1951}, + Bdsk-Url-1 = {https://projecteuclid.org:443/euclid.aoms/1177729694}, + Bdsk-Url-2 = {https://doi.org/10.1214/aoms/1177729694}} + +@book{nielsen_chuang_2010, place={Cambridge}, title={Quantum Computation and Quantum Information: 10th Anniversary Edition}, DOI={10.1017/CBO9780511976667}, publisher={Cambridge University Press}, author={Nielsen, Michael A. and Chuang, Isaac L.}, year={2010}} + + +@article{annurev-conmatphys-031119-050651, +author = {Torlai, Giacomo and Melko, Roger G.}, +title = {Machine-Learning Quantum States in the NISQ Era}, +journal = {Annual Review of Condensed Matter Physics}, +volume = {11}, +number = {1}, +pages = {325-344}, +year = {2020}, +doi = {10.1146/annurev-conmatphys-031119-050651}, + +URL = { + https://doi.org/10.1146/annurev-conmatphys-031119-050651 + +}, +eprint = { + https://doi.org/10.1146/annurev-conmatphys-031119-050651 + +} +, + abstract = { We review the development of generative modeling techniques in machine learning for the purpose of reconstructing real, noisy, many-qubit quantum states. Motivated by its interpretability and utility, we discuss in detail the theory of the restricted Boltzmann machine. We demonstrate its practical use for state reconstruction, starting from a classical thermal distribution of Ising spins, then moving systematically through increasingly complex pure and mixed quantum states. We review recent techniques in reconstruction of a cold atom wavefunction, intended for use on experimental noisy intermediate-scale quantum (NISQ) devices. Finally, we discuss the outlook for future experimental state reconstruction using machine learning in the NISQ era and beyond. } +} + + + + +@article{LeCun2015, +abstract = {Deep learning}, +author = {LeCun, Yann and Bengio, Yoshua and Hinton, Geoffrey}, +doi = {10.1038/nature14539}, +file = {:Users/btimar/Library/Application Support/Mendeley Desktop/Downloaded/LeCun, Bengio, Hinton - 2015 - Deep learning.pdf:pdf}, +issn = {0028-0836}, +journal = {Nature}, +keywords = {Computer science,Mathematics and computing}, +month = {may}, +number = {7553}, +pages = {436--444}, +publisher = {Nature Publishing Group}, +title = {{Deep learning}}, +url = {http://www.nature.com/articles/nature14539}, +volume = {521}, +year = {2015} +} + + + + +@article{Little78, +title = "Analytic study of the memory storage capacity of a neural network", +journal = "Mathematical Biosciences", +volume = "39", +number = "3", +pages = "281 - 290", +year = "1978", +issn = "0025-5564", +doi = "https://doi.org/10.1016/0025-5564(78)90058-5", +url = "http://www.sciencedirect.com/science/article/pii/0025556478900585", +author = "W.A. Little and Gordon L. Shaw", +abstract = "Previously, we developed a model of short and long term memory which was based on an analogy to the Ising spin system in a neural network. We assumed that the modification of the synaptic strengths was dependent upon the correlation of pre- and post-synaptic neuronal firing. This assumption we denote as the Hebb hypothesis. In this paper, we solve exactly a linearized version of the model and explicitly show that the capacity of the memory is related to the number of synapses rather than the much smaller number of neurons. In addition, we show that in order to utilize this large capacity, the network must store the major part of the information in the capability to generate patterns which evolve with time. We are also led to a modified Hebb hypothesis." +} + +@article{Little74, +title = "The existence of persistent states in the brain", +journal = "Mathematical Biosciences", +volume = "19", +number = "1", +pages = "101 - 120", +year = "1974", +issn = "0025-5564", +doi = "https://doi.org/10.1016/0025-5564(74)90031-5", +url = "http://www.sciencedirect.com/science/article/pii/0025556474900315", +author = "W.A. Little", +abstract = "We show that given certain plausible assumptions the existence of persistent states in a neural network can occur only if a certain transfer matrix has degenerate maximum eigenvalues. The existence of such states of persistent order is directly analogous to the existence of long range order in an Ising spin system; while the transition to the state of persistent order is analogous to the transition to the ordered phase of the spin system. It is shown that the persistent state is also characterized by correlations between neurons throughout the brain. It is suggested that these persistent states are associated with short term memory while the eigenvectors of the transfer matrix are a representation of long term memory. A numerical example is given that illustrates certain of these features." +} + + +@article{niu_universal_2019, + title = {Universal Quantum Control through Deep Reinforcement Learning}, + author = {Niu, Murphy Yuezhen and Boixo, Sergio and Smelyanskiy, Vadim N. and Neven, Hartmut}, + year = {2019}, + month = apr, + volume = {5}, + pages = {1--8}, + issn = {2056-6387}, + doi = {10.1038/s41534-019-0141-3}, + abstract = {Emerging reinforcement learning techniques using deep neural networks have shown great promise in control optimization. They harness non-local regularities of noisy control trajectories and facilitate transfer learning between tasks. To leverage these powerful capabilities for quantum control optimization, we propose a new control framework to simultaneously optimize the speed and fidelity of quantum computation against both leakage and stochastic control errors. For a broad family of two-qubit unitary gates that are important for quantum simulation of many-electron systems, we improve the control robustness by adding control noise into training environments for reinforcement learning agents trained with trusted-region-policy-optimization. The agent control solutions demonstrate a two-order-of-magnitude reduction in average-gate-error over baseline stochastic-gradient-descent solutions and up to a one-order-of-magnitude reduction in gate time from optimal gate synthesis counterparts. These significant improvements in both fidelity and runtime are achieved by combining new physical understandings and state-of-the-art machine learning techniques. Our results open a venue for wider applications in quantum simulation, quantum chemistry and quantum supremacy tests using near-term quantum devices.}, + copyright = {2019 The Author(s)}, + journal = {npj Quantum Information}, + number = {1} +} + + + + +@article{coopmans2020, + title = {Protocol {{Discovery}} for the {{Quantum Control}} of {{Majoranas}} by {{Differential Programming}} and {{Natural Evolution Strategies}}}, + author = {Coopmans, Luuk and Luo, Di and Kells, Graham and Clark, Bryan K. and Carrasquilla, Juan}, + year = {2020}, + month = aug, + abstract = {Quantum control, which refers to the active manipulation of physical systems described by the laws of quantum mechanics, constitutes an essential ingredient for the development of quantum technology. Here we apply Differentiable Programming (DP) and Natural Evolution Strategies (NES) to the optimal transport of Majorana zero modes in large superconducting nano-wires, a key element to the success of Majorana-based topological quantum computation. We formulate the motion control of Majorana fermions as an optimization problem for which we propose a new categorization of four different regimes with respect to the critical velocity of the system and the total transport time. In addition to correctly recovering the anticipated smooth protocols in the adiabatic regime, our algorithms uncover efficient but strikingly counter-intuitive motion strategies in the non-adiabatic regime. The emergent picture reveals a simple but high fidelity strategy that makes use of pulse-like jumps at the beginning and the end of the protocol with a period of constant velocity in between the jumps, which we dub the jump-move-jump protocol. We provide a transparent semi-analytical picture, which uses the sudden approximation and a reformulation of the Majorana motion in a moving frame, to illuminate the key characteristics of the jump-move-jump control strategy. Our results demonstrate that machine learning for quantum control can be applied efficiently to quantum many-body dynamical systems with performance levels that make it relevant to the realization of large-scale quantum technology.}, + archivePrefix = {arXiv}, + eprint = {2008.09128}, + eprinttype = {arxiv}, + journal = {arXiv:2008.09128 [cond-mat, physics:physics, physics:quant-ph]}, + keywords = {Condensed Matter - Strongly Correlated Electrons,Physics - Computational Physics,Quantum Physics}, + primaryClass = {cond-mat, physics:physics, physics:quant-ph} +} + + + + +@article{bukov2018, + title = {Reinforcement {{Learning}} in {{Different Phases}} of {{Quantum Control}}}, + author = {Bukov, Marin and Day, Alexandre G. R. and Sels, Dries and Weinberg, Phillip and Polkovnikov, Anatoli and Mehta, Pankaj}, + year = {2018}, + month = sep, + volume = {8}, + pages = {031086}, + doi = {10.1103/PhysRevX.8.031086}, + abstract = {The ability to prepare a physical system in a desired quantum state is central to many areas of physics such as nuclear magnetic resonance, cold atoms, and quantum computing. Yet, preparing states quickly and with high fidelity remains a formidable challenge. In this work, we implement cutting-edge reinforcement learning (RL) techniques and show that their performance is comparable to optimal control methods in the task of finding short, high-fidelity driving protocol from an initial to a target state in nonintegrable many-body quantum systems of interacting qubits. RL methods learn about the underlying physical system solely through a single scalar reward (the fidelity of the resulting state) calculated from numerical simulations of the physical system. We further show that quantum-state manipulation viewed as an optimization problem exhibits a spin-glass-like phase transition in the space of protocols as a function of the protocol duration. Our RL-aided approach helps identify variational protocols with nearly optimal fidelity, even in the glassy phase, where optimal state manipulation is exponentially hard. This study highlights the potential usefulness of RL for applications in out-of-equilibrium quantum physics.}, + journal = {Physical Review X}, + number = {3} +} + + + +@book{PDP, + editor = {Rumelhart, David E. and McClelland, James L.}, + title = {Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Vol. 1: Foundations}, + year = {1986}, + isbn = {0-262-68053-X}, + source = {softcover, \$21.95}, + publisher = {MIT Press}, + address = {Cambridge, MA, USA}, +} + + +@article {Hopfield82, + author = {Hopfield, J J}, + title = {Neural networks and physical systems with emergent collective computational abilities}, + volume = {79}, + number = {8}, + pages = {2554--2558}, + year = {1982}, + doi = {10.1073/pnas.79.8.2554}, + publisher = {National Academy of Sciences}, + abstract = {Computational properties of use of biological organisms or to the construction of computers can emerge as collective properties of systems having a large number of simple equivalent components (or neurons). The physical meaning of content-addressable memory is described by an appropriate phase space flow of the state of a system. A model of such a system is given, based on aspects of neurobiology but readily adapted to integrated circuits. The collective properties of this model produce a content-addressable memory which correctly yields an entire memory from any subpart of sufficient size. The algorithm for the time evolution of the state of the system is based on asynchronous parallel processing. Additional emergent collective properties include some capacity for generalization, familiarity recognition, categorization, error correction, and time sequence retention. The collective properties are only weakly sensitive to details of the modeling or the failure of individual devices.}, + issn = {0027-8424}, + journal = {Proceedings of the National Academy of Sciences} +} + +@article{HintonBackProp, + Author = {Rumelhart, David E. and Hinton, Geoffrey E. and Williams, Ronald J.}, + Date = {1986/10/09/online}, + Date-Added = {2018-09-17 04:18:45 +0000}, + Date-Modified = {2018-09-17 04:18:45 +0000}, + Day = {09}, + Journal = {Nature}, + L3 = {10.1038/323533a0; }, + Month = {10}, + Pages = {533 EP -}, + Publisher = {Nature Publishing Group SN -}, + Title = {Learning representations by back-propagating errors}, + Ty = {JOUR}, + Url = {http://dx.doi.org/10.1038/323533a0}, + Volume = {323}, + Year = {1986}, + Bdsk-Url-1 = {http://dx.doi.org/10.1038/323533a0}} + + +@article{Ackley85, +title = "A learning algorithm for boltzmann machines", +journal = "Cognitive Science", +volume = "9", +number = "1", +pages = "147 - 169", +year = "1985", +issn = "0364-0213", +doi = "https://doi.org/10.1016/S0364-0213(85)80012-4", +url = "http://www.sciencedirect.com/science/article/pii/S0364021385800124", +author = "David H. Ackley and Geoffrey E. Hinton and Terrence J. Sejnowski", +abstract = "The computational power of massively parallel networks of simple processing elements resides in the communication bandwidth provided by the hardware connections between elements. These connections can allow a significant fraction of the knowledge of the system to be applied to an instance of a problem in a very short time. One kind of computation for which massively parallel networks appear to be well suited is large constraint satisfaction searches, but to use the connections efficiently two conditions must be met: First, a search technique that is suitable for parallel networks must be found. Second, there must be some way of choosing internal representations which allow the preexisting hardware connections to be used efficiently for encoding the constraints in the domain being searched. We describe a general parallel search method, based on statistical mechanics, and we show how it leads to a general learning rule for modifying the connection strengths so as to incorporate knowledge about a task domain in an efficient way. We describe some simple examples in which the learning algorithm creates internal representations that are demonstrably the most efficient way of using the preexisting connectivity structure." +} + + + + + + + + + +@article{McCullogh43, + Author = {McCulloch, Warren S. and Pitts, Walter}, + Da = {1943/12/01}, + Date-Added = {2018-09-01 22:18:47 +0000}, + Date-Modified = {2018-09-01 22:18:47 +0000}, + Doi = {10.1007/BF02478259}, + Id = {McCulloch1943}, + Isbn = {1522-9602}, + Journal = {The bulletin of mathematical biophysics}, + Number = {4}, + Pages = {115--133}, + Title = {A logical calculus of the ideas immanent in nervous activity}, + Ty = {JOUR}, + Url = {https://doi.org/10.1007/BF02478259}, + Volume = {5}, + Year = {1943}, + Bdsk-Url-1 = {https://doi.org/10.1007/BF02478259}, + Bdsk-Url-2 = {http://dx.doi.org/10.1007/BF02478259}} + +@article {Melnikov1221, + author = {Melnikov, Alexey A. and Poulsen Nautrup, Hendrik and Krenn, Mario and Dunjko, Vedran and Tiersch, Markus and Zeilinger, Anton and Briegel, Hans J.}, + title = {Active learning machine learns to create new quantum experiments}, + volume = {115}, + number = {6}, + pages = {1221--1226}, + year = {2018}, + doi = {10.1073/pnas.1714936115}, + publisher = {National Academy of Sciences}, + abstract = {Quantum experiments push the envelope of our understanding of fundamental concepts in quantum physics. Modern experiments have exhaustively probed the basic notions of quantum theory. Arguably, further breakthroughs require the tackling of complex quantum phenomena and consequently require complex experiments and involved techniques. The designing of such complex experiments is difficult and often clashes with human intuition. We present an autonomous learning model which learns to design such complex experiments, without relying on previous knowledge or often flawed intuition. Our system not only learns how to design desired experiments more efficiently than the best previous approaches, but in the process also discovers nontrivial experimental techniques. Our work demonstrates that learning machines can offer dramatic advances in how experiments are generated.How useful can machine learning be in a quantum laboratory? Here we raise the question of the potential of intelligent machines in the context of scientific research. A major motivation for the present work is the unknown reachability of various entanglement classes in quantum experiments. We investigate this question by using the projective simulation model, a physics-oriented approach to artificial intelligence. In our approach, the projective simulation system is challenged to design complex photonic quantum experiments that produce high-dimensional entangled multiphoton states, which are of high interest in modern quantum experiments. The artificial intelligence system learns to create a variety of entangled states and improves the efficiency of their realization. In the process, the system autonomously (re)discovers experimental techniques which are only now becoming standard in modern quantum optical experiments{\textemdash}a trait which was not explicitly demanded from the system but emerged through the process of learning. Such features highlight the possibility that machines could have a significantly more creative role in future research.}, + issn = {0027-8424}, + URL = {https://www.pnas.org/content/115/6/1221}, + eprint = {https://www.pnas.org/content/115/6/1221.full.pdf}, + journal = {Proceedings of the National Academy of Sciences} +} + + + +@article{Dunjko_2018, + doi = {10.1088/1361-6633/aab406}, + url = {https://doi.org/10.1088%2F1361-6633%2Faab406}, + year = 2018, + month = {jun}, + publisher = {{IOP} Publishing}, + volume = {81}, + number = {7}, + pages = {074001}, + author = {Vedran Dunjko and Hans J Briegel}, + title = {Machine learning {\&} artificial intelligence in the quantum domain: a review of recent progress}, + journal = {Reports on Progress in Physics}, + abstract = {Quantum information technologies, on the one hand, and intelligent learning systems, on the other, are both emergent technologies that are likely to have a transformative impact on our society in the future. The respective underlying fields of basic research—quantum information versus machine learning (ML) and artificial intelligence (AI)—have their own specific questions and challenges, which have hitherto been investigated largely independently. However, in a growing body of recent work, researchers have been probing the question of the extent to which these fields can indeed learn and benefit from each other. Quantum ML explores the interaction between quantum computing and ML, investigating how results and techniques from one field can be used to solve the problems of the other. Recently we have witnessed significant breakthroughs in both directions of influence. For instance, quantum computing is finding a vital application in providing speed-ups for ML problems, critical in our ‘big data’ world. Conversely, ML already permeates many cutting-edge technologies and may become instrumental in advanced quantum technologies. Aside from quantum speed-up in data analysis, or classical ML optimization used in quantum experiments, quantum enhancements have also been (theoretically) demonstrated for interactive learning tasks, highlighting the potential of quantum-enhanced learning agents. Finally, works exploring the use of AI for the very design of quantum experiments and for performing parts of genuine research autonomously, have reported their first successes. Beyond the topics of mutual enhancement—exploring what ML/AI can do for quantum physics and vice versa—researchers have also broached the fundamental issue of quantum generalizations of learning and AI concepts. This deals with questions of the very meaning of learning and intelligence in a world that is fully described by quantum mechanics. In this review, we describe the main ideas, recent developments and progress in a broad spectrum of research investigating ML and AI in the quantum domain.} +} + + +@article{Rosenblatt58, + added-at = {2009-10-27T06:49:28.000+0100}, + author = {Rosenblatt, Frank}, + biburl = {https://www.bibsonomy.org/bibtex/2afcce29c2471aaa8480d2c2159361e88/chrmina}, + journal = {Psychological Review}, + keywords = {imported}, + number = 6, + pages = {386-408}, + timestamp = {2009-10-27T06:49:30.000+0100}, + title = {The Perceptron: A Probabilistic Model for Information Storage and Organization in the Brain}, + volume = 65, + year = 1958 +} + + +@article{doi:10.1080/23746149.2020.1797528, +author = { Juan Carrasquilla }, +title = {Machine learning for quantum matter}, +journal = {Advances in Physics: X}, +volume = {5}, +number = {1}, +pages = {1797528}, +year = {2020}, +publisher = {Taylor & Francis}, +doi = {10.1080/23746149.2020.1797528}, + +URL = { + https://doi.org/10.1080/23746149.2020.1797528 + +}, +eprint = { + https://doi.org/10.1080/23746149.2020.1797528 + +} + +} + + + +@book{10.5555/551283, +author = {Sutton, Richard S. and Barto, Andrew G.}, +title = {Introduction to Reinforcement Learning}, +year = {1998}, +isbn = {0262193981}, +publisher = {MIT Press}, +address = {Cambridge, MA, USA}, +edition = {1st} +} +@book{10.5555/1162264, +author = {Bishop, Christopher M.}, +title = {Pattern Recognition and Machine Learning (Information Science and Statistics)}, +year = {2006}, +isbn = {0387310738}, +publisher = {Springer-Verlag}, +address = {Berlin, Heidelberg} +} + + +@article{Samajdar2020, + title = {Complex Density Wave Orders and Quantum Phase Transitions in a Model of Square-Lattice Rydberg Atom Arrays}, + author = {Samajdar, Rhine and Ho, Wen Wei and Pichler, Hannes and Lukin, Mikhail D. and Sachdev, Subir}, + journal = {Phys. Rev. Lett.}, + volume = {124}, + issue = {10}, + pages = {103601}, + numpages = {7}, + year = {2020}, + month = {Mar}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.124.103601}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.124.103601} +} + + +@article{RevModPhys.91.045002, + title = {Machine learning and the physical sciences}, + author = {Carleo, Giuseppe and Cirac, Ignacio and Cranmer, Kyle and Daudet, Laurent and Schuld, Maria and Tishby, Naftali and Vogt-Maranto, Leslie and Zdeborov\'a, Lenka}, + journal = {Rev. Mod. Phys.}, + volume = {91}, + issue = {4}, + pages = {045002}, + numpages = {39}, + year = {2019}, + month = {Dec}, + publisher = {American Physical Society}, + doi = {10.1103/RevModPhys.91.045002}, + url = {https://link.aps.org/doi/10.1103/RevModPhys.91.045002} +} + + + +@article{SCHOLLWOCK201196, +title = "The density-matrix renormalization group in the age of matrix product states", +journal = "Annals of Physics", +volume = "326", +number = "1", +pages = "96 - 192", +year = "2011", +note = "January 2011 Special Issue", +issn = "0003-4916", +doi = "https://doi.org/10.1016/j.aop.2010.09.012", +url = "http://www.sciencedirect.com/science/article/pii/S0003491610001752", +author = "Ulrich Schollwoeck", +abstract = "The density-matrix renormalization group method (DMRG) has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. In the further development of the method, the realization that DMRG operates on a highly interesting class of quantum states, so-called matrix product states (MPS), has allowed a much deeper understanding of the inner structure of the DMRG method, its further potential and its limitations. In this paper, I want to give a detailed exposition of current DMRG thinking in the MPS language in order to make the advisable implementation of the family of DMRG algorithms in exclusively MPS terms transparent. I then move on to discuss some directions of potentially fruitful further algorithmic development: while DMRG is a very mature method by now, I still see potential for further improvements, as exemplified by a number of recently introduced algorithms." +} + + + +@article{PhysRevB.48.10345, + title = {Density-matrix algorithms for quantum renormalization groups}, + author = {White, Steven R.}, + journal = {Phys. Rev. B}, + volume = {48}, + issue = {14}, + pages = {10345--10356}, + numpages = {0}, + year = {1993}, + month = {Oct}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.48.10345}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.48.10345} +} + +@incollection{Smolensky1986, + author = {Smolensky, P.}, + chapter = {Information Processing in Dynamical Systems: Foundations of Harmony Theory}, + title = {Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Vol. 1}, + editor = {Rumelhart, David E. and McClelland, James L. and PDP Research Group, CORPORATE}, + booktitle={Parallel Distributed Processing}, + year = {1986}, + isbn = {0-262-68053-X}, + pages = {194--281}, + numpages = {88}, + url = {http://dl.acm.org/citation.cfm?id=104279.104290}, + acmid = {104290}, + publisher = {MIT Press}, + address = {Cambridge, MA, USA}, +} + +@article{PhysRevLett.69.2863, + title = {Density matrix formulation for quantum renormalization groups}, + author = {White, Steven R.}, + journal = {Phys. Rev. Lett.}, + volume = {69}, + issue = {19}, + pages = {2863--2866}, + numpages = {0}, + year = {1992}, + month = {Nov}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.69.2863}, + url = {https://link. + aps.org/doi/10.1103/PhysRevLett.69.2863} +} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% QEC + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +@article{torlai_neural_2016, + title = {Neural Decoder for Topological Codes}, + author = {Torlai, Giacomo and Melko, Roger G.}, + journal = {Phys. Rev. Lett.}, + volume = {119}, + issue = {3}, + pages = {030501}, + numpages = {5}, + year = {2017}, + month = {Jul}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.119.030501}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.119.030501} +} + + +@article{krastanov_deep_2017, + title = {Deep {Neural} {Network} {Probabilistic} {Decoder} for {Stabilizer} {Codes}}, + volume = {7}, + issn = {2045-2322}, + url = {https://doi.org/10.1038/s41598-017-11266-1}, + doi = {10.1038/s41598-017-11266-1}, + abstract = {Neural networks can efficiently encode the probability distribution of errors in an error correcting code. Moreover, these distributions can be conditioned on the syndromes of the corresponding errors. This paves a path forward for a decoder that employs a neural network to calculate the conditional distribution, then sample from the distribution - the sample will be the predicted error for the given syndrome. We present an implementation of such an algorithm that can be applied to any stabilizer code. Testing it on the toric code, it has higher threshold than a number of known decoders thanks to naturally finding the most probable error and accounting for correlations between errors.}, + number = {1}, + journal = {Scientific Reports}, + author = {Krastanov, Stefan and Jiang, Liang}, + month = sep, + year = {2017}, + pages = {11003}, +} + +@article{Varsamopoulos_2017, + doi = {10.1088/2058-9565/aa955a}, + url = {https://doi.org/10.1088%2F2058-9565%2Faa955a}, + year = 2017, + month = {nov}, + publisher = {{IOP} Publishing}, + volume = {3}, + number = {1}, + pages = {015004}, + author = {Savvas Varsamopoulos and Ben Criger and Koen Bertels}, + title = {Decoding small surface codes with feedforward neural networks}, + journal = {Quantum Science and Technology}, + abstract = {Surface codes reach high error thresholds when decoded with known algorithms, but the decoding time will likely exceed the available time budget, especially for near-term implementations. To decrease the decoding time, we reduce the decoding problem to a classification problem that a feedforward neural network can solve. We investigate quantum error correction and fault tolerance at small code distances using neural network-based decoders, demonstrating that the neural network can generalize to inputs that were not provided during training and that they can reach similar or better decoding performance compared to previous algorithms. We conclude by discussing the time required by a feedforward neural network decoder in hardware.} +} + +@article{Baireuther2018machinelearning, + doi = {10.22331/q-2018-01-29-48}, + url = {https://doi.org/10.22331/q-2018-01-29-48}, + title = {Machine-learning-assisted correction of correlated qubit errors in a topological code}, + author = {Baireuther, Paul and O'Brien, Thomas E. and Tarasinski, Brian and Beenakker, Carlo W. J.}, + journal = {{Quantum}}, + issn = {2521-327X}, + publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, + volume = {2}, + pages = {48}, + month = jan, + year = {2018} +} + +@article{Chamberland_2018, + doi = {10.1088/2058-9565/aad1f7}, + url = {https://doi.org/10.1088%2F2058-9565%2Faad1f7}, + year = 2018, + month = {jul}, + publisher = {{IOP} Publishing}, + volume = {3}, + number = {4}, + pages = {044002}, + author = {Christopher Chamberland and Pooya Ronagh}, + title = {Deep neural decoders for near term fault-tolerant experiments}, + journal = {Quantum Science and Technology}, + abstract = {Finding efficient decoders for quantum error correcting codes adapted to realistic experimental noise in fault-tolerant devices represents a significant challenge. In this paper we introduce several decoding algorithms complemented by deep neural decoders (DND) and apply them to analyze several fault-tolerant error correction (EC) protocols such as the surface code as well as Steane and Knill EC. Our methods require no knowledge of the underlying noise model afflicting the quantum device making them appealing for real-world experiments. Our analysis is based on a full circuit-level noise model. It considers both distance-three and five codes, and is performed near the codes pseudo-threshold regime. Training DND in low noise rate regimes appears to be a challenging machine learning endeavour. We provide a detailed description of our neural network architectures and training methodology. We then discuss both the advantages and limitations of DND. Lastly, we provide a rigorous analysis of the decoding runtime of trained DND and compare our methods with anticipated gate times in future quantum devices. Given the broad applications of our decoding schemes, we believe that the methods presented in this paper could have practical applications for near term fault-tolerant experiments.} +} + +@article{Breuckmann2018scalableneural, + doi = {10.22331/q-2018-05-24-68}, + url = {https://doi.org/10.22331/q-2018-05-24-68}, + title = {Scalable {N}eural {N}etwork {D}ecoders for {H}igher {D}imensional {Q}uantum {C}odes}, + author = {Breuckmann, Nikolas P. and Ni, Xiaotong}, + journal = {{Quantum}}, + issn = {2521-327X}, + publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, + volume = {2}, + pages = {68}, + month = may, + year = {2018} +} + +@article{Nautrup2019optimizingquantum, + doi = {10.22331/q-2019-12-16-215}, + url = {https://doi.org/10.22331/q-2019-12-16-215}, + title = {Optimizing {Q}uantum {E}rror {C}orrection {C}odes with {R}einforcement {L}earning}, + author = {Nautrup, Hendrik Poulsen and Delfosse, Nicolas and Dunjko, Vedran and Briegel, Hans J. and Friis, Nicolai}, + journal = {{Quantum}}, + issn = {2521-327X}, + publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, + volume = {3}, + pages = {215}, + month = dec, + year = {2019} +} + +@article{PhysRevLett.122.200501, + title = {Neural Belief-Propagation Decoders for Quantum Error-Correcting Codes}, + author = {Liu, Ye-Hua and Poulin, David}, + journal = {Phys. Rev. Lett.}, + volume = {122}, + issue = {20}, + pages = {200501}, + numpages = {6}, + year = {2019}, + month = {May}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.122.200501}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.122.200501} +} + +@article{PhysRevA.99.052351, + title = {Advantages of versatile neural-network decoding for topological codes}, + author = {Maskara, Nishad and Kubica, Aleksander and Jochym-O'Connor, Tomas}, + journal = {Phys. Rev. A}, + volume = {99}, + issue = {5}, + pages = {052351}, + numpages = {13}, + year = {2019}, + month = {May}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.99.052351}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.99.052351} +} + +@article{Andreasson2019quantumerror, + doi = {10.22331/q-2019-09-02-183}, + url = {https://doi.org/10.22331/q-2019-09-02-183}, + title = {Quantum error correction for the toric code using deep reinforcement learning}, + author = {Andreasson, Philip and Johansson, Joel and Liljestrand, Simon and Granath, Mats}, + journal = {{Quantum}}, + issn = {2521-327X}, + publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, + volume = {3}, + pages = {183}, + month = sep, + year = {2019} +} + +@article{Evert2020QEC, + author={Ryan Sweke and Markus Kesselring and Evert van Nieuwenburg and J. Eisert}, + title={Reinforcement Learning Decoders for Fault-Tolerant Quantum Computation}, + journal={Machine Learning: Science and Technology}, + url={http://iopscience.iop.org/article/10.1088/2632-2153/abc609}, + year={2020}, + abstract={Topological error correcting codes, and particularly the surface code, currently provide the most feasible road-map towards large-scale fault-tolerant quantum computation. As such, obtaining fast and flexible decoding algorithms for these codes, within the experimentally realistic and challenging context of faulty syndrome measurements, without requiring any final read-out of the physical qubits, is of critical importance. In this work, we show that the problem of decoding such codes can be naturally reformulated as a process of repeated interactions between a decoding agent and a code environment, to which the machinery of reinforcement learning can be applied to obtain decoding agents. While in principle this framework can be instantiated with environments modelling circuit level noise, we take a first step towards this goal by using deepQ learning to obtain decoding agents for a variety of simplified phenomenological noise models, which yield faulty syndrome measurements without including the propagation of errors which arise in full circuit level noise models.} +} + +@article{Ni2020neuralnetwork, + doi = {10.22331/q-2020-08-24-310}, + url = {https://doi.org/10.22331/q-2020-08-24-310}, + title = {Neural {N}etwork {D}ecoders for {L}arge-{D}istance 2{D} {T}oric {C}odes}, + author = {Ni, Xiaotong}, + journal = {{Quantum}}, + issn = {2521-327X}, + publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, + volume = {4}, + pages = {310}, + month = aug, + year = {2020} +} + +@article{PhysRevResearch.2.033399, + title = {General framework for constructing fast and near-optimal machine-learning-based decoder of the topological stabilizer codes}, + author = {Davaasuren, Amarsanaa and Suzuki, Yasunari and Fujii, Keisuke and Koashi, Masato}, + journal = {Phys. Rev. Research}, + volume = {2}, + issue = {3}, + pages = {033399}, + numpages = {25}, + year = {2020}, + month = {Sep}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevResearch.2.033399}, + url = {https://link.aps.org/doi/10.1103/PhysRevResearch.2.033399} +} + +@article{PhysRevResearch.2.023230, + title = {Deep Q-learning decoder for depolarizing noise on the toric code}, + author = {Fitzek, David and Eliasson, Mattias and Kockum, Anton Frisk and Granath, Mats}, + journal = {Phys. Rev. Research}, + volume = {2}, + issue = {2}, + pages = {023230}, + numpages = {17}, + year = {2020}, + month = {May}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevResearch.2.023230}, + url = {https://link.aps.org/doi/10.1103/PhysRevResearch.2.023230} +} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% QUANTUM TOMOGRAPHY + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +@article{PhysRevA.55.R1561, + title = {Quantum-state estimation}, + author = {Hradil, Z.}, + journal = {Phys. Rev. A}, + volume = {55}, + issue = {3}, + pages = {R1561--R1564}, + numpages = {0}, + year = {1997}, + month = {Mar}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.55.R1561}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.55.R1561} +} + +@article{PhysRevA.63.040303, + title = {Iterative algorithm for reconstruction of entangled states}, + author = {\ifmmode \check{R}\else \v{R}\fi{}eh\'a\ifmmode \check{c}\else \v{c}\fi{}ek, J. and Hradil, Z. and Je\ifmmode \check{z}\else \v{z}\fi{}ek, M.}, + journal = {Phys. Rev. A}, + volume = {63}, + issue = {4}, + pages = {040303}, + numpages = {4}, + year = {2001}, + month = {Mar}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.63.040303}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.63.040303} +} + + +@article{Jezek2003, +author = {Je{\v{z}}ek, Miroslav and Fiur{\'{a}}{\v{s}}ek, Jarom{\'{i}}r and Hradil, Zden{\v{e}}k}, +doi = {10.1103/PhysRevA.68.012305}, +issn = {1050-2947}, +journal = {Physical Review A}, +month = {jul}, +number = {1}, +pages = {012305}, +publisher = {American Physical Society}, +title = {{Quantum inference of states and processes}}, +url = {https://link.aps.org/doi/10.1103/PhysRevA.68.012305}, +volume = {68}, +year = {2003} +} + +@article{PhysRevA.64.052312, + title = {Measurement of qubits}, + author = {James, Daniel F. V. and Kwiat, Paul G. and Munro, William J. and White, Andrew G.}, + journal = {Phys. Rev. A}, + volume = {64}, + issue = {5}, + pages = {052312}, + numpages = {15}, + year = {2001}, + month = {Oct}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.64.052312}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.64.052312} +} + + +@article{Banaszek2013, +author = {Banaszek, K and Cramer, M and Gross, D}, +doi = {10.1088/1367-2630/15/12/125020}, +issn = {1367-2630}, +journal = {New Journal of Physics}, +month = {dec}, +number = {12}, +pages = {125020}, +title = {{Focus on quantum tomography}}, +url = {http://stacks.iop.org/1367-2630/15/i=12/a=125020?key=crossref.d640bfdf036841021cae0e5aafd08554}, +volume = {15}, +year = {2013} +} + + +@article{vogel89, + title = {Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase}, + author = {Vogel, K. and Risken, H.}, + journal = {Phys. Rev. A}, + volume = {40}, + issue = {5}, + pages = {2847--2849}, + numpages = {0}, + year = {1989}, + month = {Sep}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.40.2847}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.40.2847} +} + + +@article{PhysRevLett.105.150401, + title = {Quantum State Tomography via Compressed Sensing}, + author = {Gross, David and Liu, Yi-Kai and Flammia, Steven T. and Becker, Stephen and Eisert, Jens}, + journal = {Phys. Rev. Lett.}, + volume = {105}, + issue = {15}, + pages = {150401}, + numpages = {4}, + year = {2010}, + month = {Oct}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.105.150401}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.105.150401} +} + +@article{Flammia_2012, + doi = {10.1088/1367-2630/14/9/095022}, + url = {https://doi.org/10.1088%2F1367-2630%2F14%2F9%2F095022}, + year = 2012, + month = {sep}, + publisher = {{IOP} Publishing}, + volume = {14}, + number = {9}, + pages = {095022}, + author = {Steven T Flammia and David Gross and Yi-Kai Liu and Jens Eisert}, + title = {Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators}, + journal = {New Journal of Physics}, + abstract = {Intuitively, if a density operator has small rank, then it should be easier to estimate from experimental data, since in this case only a few eigenvectors need to be learned. We prove two complementary results that confirm this intuition. Firstly, we show that a low-rank density matrix can be estimated using fewer copies of the state, i.e. the sample complexity of tomography decreases with the rank. Secondly, we show that unknown low-rank states can be reconstructed from an incomplete set of measurements, using techniques from compressed sensing and matrix completion. These techniques use simple Pauli measurements, and their output can be certified without making any assumptions about the unknown state. In this paper, we present a new theoretical analysis of compressed tomography, based on the restricted isometry property for low-rank matrices. Using these tools, we obtain near-optimal error bounds for the realistic situation where the data contain noise due to finite statistics, and the density matrix is full-rank with decaying eigenvalues. We also obtain upper bounds on the sample complexity of compressed tomography, and almost-matching lower bounds on the sample complexity of any procedure using adaptive sequences of Pauli measurements. Using numerical simulations, we compare the performance of two compressed sensing estimators—the matrix Dantzig selector and the matrix Lasso—with standard maximum-likelihood estimation (MLE). We find that, given comparable experimental resources, the compressed sensing estimators consistently produce higher fidelity state reconstructions than MLE. In addition, the use of an incomplete set of measurements leads to faster classical processing with no loss of accuracy. Finally, we show how to certify the accuracy of a low-rank estimate using direct fidelity estimation, and describe a method for compressed quantum process tomography that works for processes with small Kraus rank and requires only Pauli eigenstate preparations and Pauli measurements.} +} + + +@article{riofrio_experimental_2017, + title = {Experimental quantum compressed sensing for a seven-qubit system}, + volume = {8}, + issn = {2041-1723}, + url = {https://doi.org/10.1038/ncomms15305}, + doi = {10.1038/ncomms15305}, + abstract = {Well-controlled quantum devices with their increasing system size face a new roadblock hindering further development of quantum technologies. The effort of quantum tomography—the reconstruction of states and processes of a quantum device—scales unfavourably: state-of-the-art systems can no longer be characterized. Quantum compressed sensing mitigates this problem by reconstructing states from incomplete data. Here we present an experimental implementation of compressed tomography of a seven-qubit system—a topological colour code prepared in a trapped ion architecture. We are in the highly incomplete—127 Pauli basis measurement settings—and highly noisy—100 repetitions each—regime. Originally, compressed sensing was advocated for states with few non-zero eigenvalues. We argue that low-rank estimates are appropriate in general since statistical noise enables reliable reconstruction of only the leading eigenvectors. The remaining eigenvectors behave consistently with a random-matrix model that carries no information about the true state.}, + number = {1}, + journal = {Nature Communications}, + author = {Riofrío, C. A. and Gross, D. and Flammia, S. T. and Monz, T. and Nigg, D. and Blatt, R. and Eisert, J.}, + month = may, + year = {2017}, + pages = {15305}, +} + + +@article{Blume_Kohout_2010, + doi = {10.1088/1367-2630/12/4/043034}, + url = {https://doi.org/10.1088%2F1367-2630%2F12%2F4%2F043034}, + year = 2010, + month = {apr}, + publisher = {{IOP} Publishing}, + volume = {12}, + number = {4}, + pages = {043034}, + author = {Robin Blume-Kohout}, + title = {Optimal, reliable estimation of quantum states}, + journal = {New Journal of Physics}, + abstract = {Accurately inferring the state of a quantum device from the results of measurements is a crucial task in building quantum information processing hardware. The predominant state estimation procedure, maximum likelihood estimation (MLE), generally reports an estimate with zero eigenvalues. These cannot be justified. Furthermore, the MLE estimate is incompatible with error bars, so conclusions drawn from it are suspect. I propose an alternative procedure, Bayesian mean estimation (BME). BME never yields zero eigenvalues, its eigenvalues provide a bound on their own uncertainties, and under certain circumstances it is provably the most accurate procedure possible. I show how to implement BME numerically, and how to obtain natural error bars that are compatible with the estimate. Finally, I briefly discuss the differences between Bayesian and frequentist estimation techniques.} +} +@article{Granade_2017, + doi = {10.1088/1367-2630/aa8fe6}, + url = {https://doi.org/10.1088%2F1367-2630%2Faa8fe6}, + year = 2017, + month = {nov}, + publisher = {{IOP} Publishing}, + volume = {19}, + number = {11}, + pages = {113017}, + author = {Christopher Granade and Christopher Ferrie and Steven T Flammia}, + title = {Practical adaptive quantum tomography}, + journal = {New Journal of Physics}, + abstract = {We introduce a fast and accurate heuristic for adaptive tomography that addresses many of the limitations of prior methods. Previous approaches were either too computationally intensive or tailored to handle special cases such as single qubits or pure states. By contrast, our approach combines the efficiency of online optimization with generally applicable and well-motivated data-processing techniques. We numerically demonstrate these advantages in several scenarios including mixed states, higher-dimensional systems, and restricted measurements.} +} + + +@article{PhysRevLett.108.070502, + title = {Efficient Method for Computing the Maximum-Likelihood Quantum State from Measurements with Additive Gaussian Noise}, + author = {Smolin, John A. and Gambetta, Jay M. and Smith, Graeme}, + journal = {Phys. Rev. Lett.}, + volume = {108}, + issue = {7}, + pages = {070502}, + numpages = {4}, + year = {2012}, + month = {Feb}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.108.070502}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.108.070502} +} + +@article{PhysRevLett.106.100401, + title = {Efficient Measurement of Quantum Dynamics via Compressive Sensing}, + author = {Shabani, A. and Kosut, R. L. and Mohseni, M. and Rabitz, H. and Broome, M. A. and Almeida, M. P. and Fedrizzi, A. and White, A. G.}, + journal = {Phys. Rev. Lett.}, + volume = {106}, + issue = {10}, + pages = {100401}, + numpages = {4}, + year = {2011}, + month = {Mar}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.106.100401}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.106.100401} +} + +@article{cramer2009efficient, + Author = {Cramer, M and Plenio, MB and Flammia, ST and Somma, R and Gross, D and Bartlett, SD and Landon-Cardinal, O and Poulin, D and Liu, YK}, + Date-Added = {2020-05-19 19:53:36 +0000}, + Date-Modified = {2020-05-19 19:53:36 +0000}, + Journal = {Nature communications}, + Pages = {149}, + Title = {Efficient quantum state tomography.}, + Url = {http://www.nature.com/articles/ncomms1147}, + Volume = {1}, + Year = {2009}, + Bdsk-Url-1 = {http://www.nature.com/articles/ncomms1147}} + + +@article{MPOtomo, + Author = {Baumgratz, T. and Gross, D. and Cramer, M. and Plenio, M. B.}, + Doi = {10.1103/PhysRevLett.111.020401}, + Issue = {2}, + Journal = {Phys. Rev. Lett.}, + Month = {Jul}, + Numpages = {5}, + Pages = {020401}, + Publisher = {American Physical Society}, + Title = {Scalable Reconstruction of Density Matrices}, + Url = {https://link.aps.org/doi/10.1103/PhysRevLett.111.020401}, + Volume = {111}, + Year = {2013}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.111.020401}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.111.020401}} + +@article{LeiWang2020, + Author = {Wang, Jun and Han, Zhao-Yu and Wang, Song-Bo and Li, Zeyang and Mu, Liang-Zhu and Fan, Heng and Wang, Lei}, + Doi = {10.1103/PhysRevA.101.032321}, + Issue = {3}, + Journal = {Phys. Rev. A}, + Month = {Mar}, + Numpages = {11}, + Pages = {032321}, + Publisher = {American Physical Society}, + Title = {Scalable quantum tomography with fidelity estimation}, + Url = {https://link.aps.org/doi/10.1103/PhysRevA.101.032321}, + Volume = {101}, + Year = {2020}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.101.032321}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.101.032321}} + + +@article{huang_predicting_2020, + title = {Predicting many properties of a quantum system from very few measurements}, + volume = {16}, + issn = {1745-2481}, + url = {https://doi.org/10.1038/s41567-020-0932-7}, + doi = {10.1038/s41567-020-0932-7}, + abstract = {Predicting the properties of complex, large-scale quantum systems is essential for developing quantum technologies. We present an efficient method for constructing an approximate classical description of a quantum state using very few measurements of the state. This description, called a ‘classical shadow’, can be used to predict many different properties; order \$\$\{{\textbackslash}mathrm\{log\}\}{\textbackslash},(M)\$\$log(M)measurements suffice to accurately predict M different functions of the state with high success probability. The number of measurements is independent of the system size and saturates information-theoretic lower bounds. Moreover, target properties to predict can be selected after the measurements are completed. We support our theoretical findings with extensive numerical experiments. We apply classical shadows to predict quantum fidelities, entanglement entropies, two-point correlation functions, expectation values of local observables and the energy variance of many-body local Hamiltonians. The numerical results highlight the advantages of classical shadows relative to previously known methods.}, + number = {10}, + journal = {Nature Physics}, + author = {Huang, Hsin-Yuan and Kueng, Richard and Preskill, John}, + month = oct, + year = {2020}, + pages = {1050--1057}, +} + + +@article{Lanyon2017, + Abstract = {Traditionally quantum state tomography is used to characterize a quantum state, but it becomes exponentially hard with the system size. An alternative technique, matrix product state tomography, is shown to work well in practical situations.}, + Author = {Lanyon, B. P. and Maier, C. and Holz{\"{a}}pfel, M. and Baumgratz, T. and Hempel, C. and Jurcevic, P. and Dhand, I. and Buyskikh, A. S. and Daley, A. J. and Cramer, M. and Plenio, M. B. and Blatt, R. and Roos, C. F.}, + Doi = {10.1038/nphys4244}, + File = {:Users/btimar/Library/Application Support/Mendeley Desktop/Downloaded/Lanyon et al. - 2017 - Efficient tomography of a quantum many-body system.pdf:pdf}, + Issn = {1745-2473}, + Journal = {Nature Physics}, + Keywords = {Quantum information,Quantum simulation}, + Month = {sep}, + Number = {12}, + Pages = {1158--1162}, + Publisher = {Nature Publishing Group}, + Title = {{Efficient tomography of a quantum many-body system}}, + Url = {http://www.nature.com/doifinder/10.1038/nphys4244}, + Volume = {13}, + Year = {2017}, + Bdsk-Url-1 = {http://www.nature.com/doifinder/10.1038/nphys4244}, + Bdsk-Url-2 = {https://doi.org/10.1038/nphys4244}} + +@article{eisert_quantum_2020, + title = {Quantum certification and benchmarking}, + volume = {2}, + issn = {2522-5820}, + url = {https://doi.org/10.1038/s42254-020-0186-4}, + doi = {10.1038/s42254-020-0186-4}, + abstract = {With the rapid development of quantum technologies, a pressing need has emerged for a wide array of tools for the certification and characterization of quantum devices. Such tools are critical because the powerful applications of quantum information science will only be realized if stringent levels of precision of components can be reached and their functioning guaranteed. This Technical Review provides a brief overview of the known characterization methods for certification, benchmarking and tomographic reconstruction of quantum states and processes, and outlines their applications in quantum computing, simulation and communication.}, + number = {7}, + journal = {Nature Reviews Physics}, + author = {Eisert, Jens and Hangleiter, Dominik and Walk, Nathan and Roth, Ingo and Markham, Damian and Parekh, Rhea and Chabaud, Ulysse and Kashefi, Elham}, + month = jul, + year = {2020}, + pages = {382--390}, +} + + + + + +@misc{pastaq, + title={\mbox{PastaQ}: A Package for Simulation, Tomography and Analysis of Quantum Computers}, + author={Matthew Fishman and Giacomo Torlai}, + year={2020}, + url={https://github.com/GTorlai/PastaQ.jl/} +} + + + +@ARTICLE{qucumber, + author = {{Beach}, Matthew J.~S. and {De Vlugt}, Isaac and {Golubeva}, Anna and + {Huembeli}, Patrick and {Kulchytskyy}, Bohdan and {Luo}, Xiuzhe and + {Melko}, Roger G. and {Merali}, Ejaaz and {Torlai}, Giacomo}, + title = "{QuCumber: wavefunction reconstruction with neural networks}", + journal = {arXiv e-prints}, + keywords = {Quantum Physics, Condensed Matter - Strongly Correlated Electrons}, + year = "2018", + month = "Dec", + eid = {arXiv:1812.09329}, + pages = {arXiv:1812.09329}, +archivePrefix = {arXiv}, + eprint = {1812.09329}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/\#abs/2018arXiv181209329B}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@article{PhysRevB.85.165146, + title = {Perfect sampling with unitary tensor networks}, + author = {Ferris, Andrew J. and Vidal, Guifre}, + journal = {Phys. Rev. B}, + volume = {85}, + issue = {16}, + pages = {165146}, + numpages = {10}, + year = {2012}, + month = {Apr}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.85.165146}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.85.165146} +} + + +@ARTICLE{netket, + author = {{Carleo}, Giuseppe and {Choo}, Kenny and {Hofmann}, Damian and + {Smith}, James E.~T. and {Westerhout}, Tom and {Alet}, Fabien and + {Davis}, Emily J. and {Efthymiou}, Stavros and {Glasser}, Ivan and + {Lin}, Sheng-Hsuan and {Mauri}, Marta and {Mazzola}, Guglielmo and + {Mendl}, Christian B. and {van Nieuwenburg}, Evert and + {O'Reilly}, Ossian and {Th{\'e}veniaut}, Hugo and {Torlai}, Giacomo and + {Wietek}, Alexander}, + title = "{NetKet: A Machine Learning Toolkit for Many-Body Quantum Systems}", + journal = {arXiv e-prints}, + keywords = {Quantum Physics, Condensed Matter - Disordered Systems and Neural Networks, Condensed Matter - Strongly Correlated Electrons, Physics - Computational Physics, Physics - Data Analysis, Statistics and Probability}, + year = "2019", + month = "Mar", + eid = {arXiv:1904.00031}, + pages = {arXiv:1904.00031}, +archivePrefix = {arXiv}, + eprint = {1904.00031}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2019arXiv190400031C}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + + + +@article{torlai_learning_2016, + Abstract = {A Boltzmann machine is a stochastic neural network that has been extensively used in the layers of deep architectures for modern machine learning applications. In this paper, we develop a Boltzmann machine that is capable of modeling thermodynamic observables for physical systems in thermal equilibrium. Through unsupervised learning, we train the Boltzmann machine on data sets constructed with spin configurations importance sampled from the partition function of an Ising Hamiltonian at different temperatures using Monte Carlo (MC) methods. The trained Boltzmann machine is then used to generate spin states, for which we compare thermodynamic observables to those computed by direct MC sampling. We demonstrate that the Boltzmann machine can faithfully reproduce the observables of the physical system. Further, we observe that the number of neurons required to obtain accurate results increases as the system is brought close to criticality.}, + Author = {Torlai, Giacomo and Melko, Roger G.}, + Doi = {10.1103/PhysRevB.94.165134}, + Journal = {Physical Review B}, + Month = oct, + Number = {16}, + Pages = {165134}, + Title = {Learning thermodynamics with {Boltzmann} machines}, + Url = {http://link.aps.org/doi/10.1103/PhysRevB.94.165134}, + Urldate = {2017-01-20}, + Volume = {94}, + Year = {2016}, + Bdsk-Url-1 = {http://link.aps.org/doi/10.1103/PhysRevB.94.165134}, + Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevB.94.165134}} + + +@article{torlai_Tomo, + Author = {Torlai, Giacomo and Mazzola, Guglielmo and Carrasquilla, Juan and Troyer, Matthias and Melko, Roger and Carleo, Giuseppe}, + Doi = {10.1038/s41567-018-0048-5}, + Journal = {Nature Physics}, + Number = {5}, + Pages = {447--450}, + Title = {Neural-network quantum state tomography}, + Url = {https://doi.org/10.1038/s41567-018-0048-5}, + Volume = {14}, + Year = {2018} +} + + +@article{rocchetto, + Abstract = {The exact description of many-body quantum systems represents one of the major challenges in modern physics, because it requires an amount of computational resources that scales exponentially with the size of the system. Simulating the evolution of a state, or even storing its description, rapidly becomes intractable for exact classical algorithms. Recently, machine learning techniques, in the form of restricted Boltzmann machines, have been proposed as a way to efficiently represent certain quantum states with applications in state tomography and ground state estimation. Here, we introduce a practically usable deep architecture for representing and sampling from probability distributions of quantum states. Our representation is based on variational auto-encoders, a type of generative model in the form of a neural network. We show that this model is able to learn efficient representations of states that are easy to simulate classically and can compress states that are not classically tractable. Specifically, we consider the learnability of a class of quantum states introduced by Fefferman and Umans. Such states are provably hard to sample for classical computers, but not for quantum ones, under plausible computational complexity assumptions. The good level of compression achieved for hard states suggests these methods can be suitable for characterizing states of the size expected in first generation quantum hardware.}, + Author = {Rocchetto, Andrea and Grant, Edward and Strelchuk, Sergii and Carleo, Giuseppe and Severini, Simone}, + Da = {2018/06/28}, + Date-Added = {2019-03-27 20:12:35 +0000}, + Date-Modified = {2019-03-27 20:12:35 +0000}, + Doi = {10.1038/s41534-018-0077-z}, + Id = {Rocchetto2018}, + Isbn = {2056-6387}, + Journal = {npj Quantum Information}, + Number = {1}, + Pages = {28}, + Title = {Learning hard quantum distributions with variational autoencoders}, + Ty = {JOUR}, + Url = {https://doi.org/10.1038/s41534-018-0077-z}, + Volume = {4}, + Year = {2018}, + Bdsk-Url-1 = {https://doi.org/10.1038/s41534-018-0077-z}} + + + +@article{Torlai_latent, + title = {Latent Space Purification via Neural Density Operators}, + author = {Torlai, Giacomo and Melko, Roger G.}, + journal = {Phys. Rev. Lett.}, + volume = {120}, + issue = {24}, + pages = {240503}, + numpages = {5}, + year = {2018}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.120.240503}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.120.240503} +} + + +@ARTICLE{2018arXiv181206693Q, + author = {{Quek}, Yihui and {Fort}, Stanislav and {Khoon Ng}, Hui}, + title = "{Adaptive Quantum State Tomography with Neural Networks}", + journal = {arXiv e-prints}, + keywords = {Quantum Physics, Computer Science - Machine Learning}, + year = 2018, + month = dec, + eid = {arXiv:1812.06693}, + pages = {arXiv:1812.06693}, +archivePrefix = {arXiv}, + eprint = {1812.06693}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2018arXiv181206693Q}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@article{carrasquilla_povm, + Abstract = {A major bottleneck in the development of scalable many-body quantum technologies is the difficulty in benchmarking state preparations, which suffer from an exponential `curse of dimensionality'inherent to the classical description of quantum states. We present an experimentally friendly method for density matrix reconstruction based on neural network generative models. The learning procedure comes with a built-in approximate certificate of the reconstruction and makes no assumptions about the purity of the state under scrutiny. It can efficiently handle a broad class of complex systems including prototypical states in quantum information, as well as ground states of local spin models common to condensed matter physics. The key insight is to reduce state tomography to an unsupervised learning problem of the statistics of an informationally complete quantum measurement. This constitutes a modern machine learning approach to the validation of complex quantum devices, which may in addition prove relevant as a neural-network ansatz over mixed states suitable for variational optimization.}, + Author = {Carrasquilla, Juan and Torlai, Giacomo and Melko, Roger G. and Aolita, Leandro}, + Da = {2019/03/01}, + Date-Added = {2019-03-28 12:52:13 +0000}, + Date-Modified = {2019-03-28 12:52:13 +0000}, + Doi = {10.1038/s42256-019-0028-1}, + Id = {Carrasquilla2019}, + Isbn = {2522-5839}, + Journal = {Nature Machine Intelligence}, + Number = {3}, + Pages = {155--161}, + Title = {Reconstructing quantum states with generative models}, + Ty = {JOUR}, + Url = {https://doi.org/10.1038/s42256-019-0028-1}, + Volume = {1}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1038/s42256-019-0028-1}} + + + +@article{biamonte_qst, + title = {Experimental neural network enhanced quantum tomography}, + volume = {6}, + issn = {2056-6387}, + url = {https://doi.org/10.1038/s41534-020-0248-6}, + doi = {10.1038/s41534-020-0248-6}, + abstract = {Quantum tomography is currently ubiquitous for testing any implementation of a quantum information processing device. Various sophisticated procedures for state and process reconstruction from measured data are well developed and benefit from precise knowledge of the model describing state-preparation-and-measurement (SPAM) apparatus. However, physical models suffer from intrinsic limitations as actual measurement operators and trial states cannot be known precisely. This scenario inevitably leads to SPAM errors degrading reconstruction performance. Here we develop a framework based on machine learning which generally applies to both the tomography and SPAM mitigation problem. We experimentally implement our method. We trained a supervised neural network to filter the experimental data and hence uncovered salient patterns that characterize the measurement probabilities for the original state and the ideal experimental apparatus free from SPAM errors. We compared the neural network state reconstruction protocol with a protocol treating SPAM errors by process tomography, as well as to an SPAM-agnostic protocol with idealized measurements. The average reconstruction fidelity is shown to be enhanced by 10\% and 27\%, respectively. The presented methods apply to the vast range of quantum experiments which rely on tomography.}, + number = {1}, + journal = {npj Quantum Information}, + author = {Palmieri, Adriano Macarone and Kovlakov, Egor and Bianchi, Federico and Yudin, Dmitry and Straupe, Stanislav and Biamonte, Jacob D. and Kulik, Sergei}, + month = feb, + year = {2020}, + pages = {20}, +} + + + + +@ARTICLE{torlai19, + author = {{Torlai}, Giacomo and {Timar}, Brian and {van Nieuwenburg}, Evert P.~L. and + {Levine}, Harry and {Omran}, Ahmed and {Keesling}, Alexander and + {Bernien}, Hannes and {Greiner}, Markus and {Vuleti{\'c}}, Vladan and + {Lukin}, Mikhail D. and {Melko}, Roger G. and {Endres}, Manuel}, + title = "{Integrating Neural Networks with a Quantum Simulator for State Reconstruction}", + journal = {arXiv e-prints}, + keywords = {Quantum Physics, Condensed Matter - Quantum Gases}, + year = "2019", + month = "Apr", + eid = {arXiv:1904.08441}, + pages = {arXiv:1904.08441}, +archivePrefix = {arXiv}, + eprint = {1904.08441}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2019arXiv190408441T}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@article{torlai_rydberg19, + title = {Integrating Neural Networks with a Quantum Simulator for State Reconstruction}, + author = {Torlai, Giacomo and Timar, Brian and van Nieuwenburg, Evert P. L. and Levine, Harry and Omran, Ahmed and Keesling, Alexander and Bernien, Hannes and Greiner, Markus and Vuleti\ifmmode \acute{c}\else \'{c}\fi{}, Vladan and Lukin, Mikhail D. and Melko, Roger G. and Endres, Manuel}, + journal = {Phys. Rev. Lett.}, + volume = {123}, + issue = {23}, + pages = {230504}, + numpages = {6}, + year = {2019}, + month = {Dec}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.123.230504}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.123.230504} +} + + +@article{xin_local-measurement-based_2019, + title = {Local-measurement-based quantum state tomography via neural networks}, + volume = {5}, + issn = {2056-6387}, + url = {https://doi.org/10.1038/s41534-019-0222-3}, + doi = {10.1038/s41534-019-0222-3}, + abstract = {Quantum state tomography is a daunting challenge of experimental quantum computing, even in moderate system size. One way to boost the efficiency of state tomography is via local measurements on reduced density matrices, but the reconstruction of the full state thereafter is hard. Here, we present a machine-learning method to recover the ground states of \$\$k\$\$k-local Hamiltonians from just the local information, where a fully connected neural network is built to fulfill the task with up to seven qubits. In particular, we test the neural network model with a practical dataset, that in a 4-qubit nuclear magnetic resonance system our method yields global states via the 2-local information with high accuracy. Our work paves the way towards scalable state tomography in large quantum systems.}, + number = {1}, + journal = {npj Quantum Information}, + author = {Xin, Tao and Lu, Sirui and Cao, Ningping and Anikeeva, Galit and Lu, Dawei and Li, Jun and Long, Guilu and Zeng, Bei}, + month = nov, + year = {2019}, + pages = {109}, +} + + + +@article{Sehayek2019, + title = {Learnability scaling of quantum states: Restricted Boltzmann machines}, + author = {Sehayek, Dan and Golubeva, Anna and Albergo, Michael S. and Kulchytskyy, Bohdan and Torlai, Giacomo and Melko, Roger G.}, + journal = {Phys. Rev. B}, + volume = {100}, + issue = {19}, + pages = {195125}, + numpages = {7}, + year = {2019}, + month = {Nov}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.100.195125}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.100.195125} +} + + +@article{torlai_chemistry, + title = {Precise measurement of quantum observables with neural-network estimators}, + author = {Torlai, Giacomo and Mazzola, Guglielmo and Carleo, Giuseppe and Mezzacapo, Antonio}, + journal = {Phys. Rev. Research}, + volume = {2}, + issue = {2}, + pages = {022060}, + numpages = {6}, + year = {2020}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevResearch.2.022060}, + url = {https://link.aps.org/doi/10.1103/PhysRevResearch.2.022060} +} + + + +@article{PhysRevA.102.022412, + title = {Eigenstate extraction with neural-network tomography}, + author = {Melkani, Abhijeet and Gneiting, Clemens and Nori, Franco}, + journal = {Phys. Rev. A}, + volume = {102}, + issue = {2}, + pages = {022412}, + numpages = {11}, + year = {2020}, + month = {Aug}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.102.022412}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.102.022412} +} + +@ARTICLE{NoriGAN, + author = {{Ahmed}, Shahnawaz and {S{\'a}nchez Mu{\~n}oz}, Carlos and {Nori}, Franco and {Frisk Kockum}, Anton}, + title = "{Quantum State Tomography with Conditional Generative Adversarial Networks}", + journal = {arXiv e-prints}, + keywords = {Quantum Physics, Computer Science - Machine Learning}, + year = 2020, + month = aug, + eid = {arXiv:2008.03240}, + pages = {arXiv:2008.03240}, +archivePrefix = {arXiv}, + eprint = {2008.03240}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv200803240A}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@article{Tiunov:20, +author = {E. S. Tiunov and V. V. Tiunova (Vyborova) and A. E. Ulanov and A. I. Lvovsky and A. K. Fedorov}, +journal = {Optica}, +keywords = {Atomic ensembles; Condensed matter; Neural networks; Optomechanics; Parametric oscillators; Quantum information processing}, +number = {5}, +pages = {448--454}, +publisher = {OSA}, +title = {Experimental quantum homodyne tomography via machine learning}, +volume = {7}, +month = {May}, +year = {2020}, +url = {http://www.osapublishing.org/optica/abstract.cfm?URI=optica-7-5-448}, +doi = {10.1364/OPTICA.389482}, +abstract = {Complete characterization of states and processes that occur within quantum devices is crucial for understanding and testing their potential to outperform classical technologies for communications and computing. However, solving this task with current state-of-the-art techniques becomes unwieldy for large and complex quantum systems. Here we realize and experimentally demonstrate a method for complete characterization of a quantum harmonic oscillator based on an artificial neural network known as the restricted Boltzmann machine. We apply the method to optical homodyne tomography and show it to allow full estimation of quantum states based on a smaller amount of experimental data compared to state-of-the-art methods. We link this advantage to reduced overfitting. Although our experiment is in the optical domain, our method provides a way of exploring quantum resources in a broad class of large-scale physical systems, such as superconducting circuits, atomic and molecular ensembles, and optomechanical systems.}, +} + +@ARTICLE{Cha2020, + author = {{Cha}, Peter and {Ginsparg}, Paul and {Wu}, Felix and {Carrasquilla}, Juan and {McMahon}, Peter L. and {Kim}, Eun-Ah}, + title = "{Attention-based Quantum Tomography}", + journal = {arXiv e-prints}, + keywords = {Quantum Physics, Computer Science - Machine Learning}, + year = 2020, + month = jun, + eid = {arXiv:2006.12469}, + pages = {arXiv:2006.12469}, +archivePrefix = {arXiv}, + eprint = {2006.12469}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv200612469C}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@article{PhysRevA.102.042604, + title = {Neural-network quantum state tomography in a two-qubit experiment}, + author = {Neugebauer, Marcel and Fischer, Laurin and J\"ager, Alexander and Czischek, Stefanie and Jochim, Selim and Weidem\"uller, Matthias and G\"arttner, Martin}, + journal = {Phys. Rev. A}, + volume = {102}, + issue = {4}, + pages = {042604}, + numpages = {7}, + year = {2020}, + month = {Oct}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.102.042604}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.102.042604} +} + +@article{DeVlugt2020, + title = {Reconstructing quantum molecular rotor ground states}, + author = {De Vlugt, Isaac J. S. and Iouchtchenko, Dmitri and Merali, Ejaaz and Roy, Pierre-Nicholas and Melko, Roger G.}, + journal = {Phys. Rev. B}, + volume = {102}, + issue = {3}, + pages = {035108}, + numpages = {10}, + year = {2020}, + month = {Jul}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.102.035108}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.102.035108} +} + + +@ARTICLE{2020arXiv200907601S, + author = {{Smith}, Alistair W.~R. and {Gray}, Johnnie and {Kim}, M.~S.}, + title = "{Efficient Approximate Quantum State Tomography with Basis Dependent Neural-Networks}", + journal = {arXiv e-prints}, + keywords = {Quantum Physics, Computer Science - Machine Learning}, + year = 2020, + month = sep, + eid = {arXiv:2009.07601}, + pages = {arXiv:2009.07601}, +archivePrefix = {arXiv}, + eprint = {2009.07601}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv200907601S}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + +@ARTICLE{torlai_QPT, + author = {{Torlai}, Giacomo and {Wood}, Christopher J. and {Acharya}, Atithi and {Carleo}, Giuseppe and {Carrasquilla}, Juan and {Aolita}, Leandro}, + title = "{Quantum process tomography with unsupervised learning and tensor networks}", + journal = {arXiv e-prints}, + keywords = {Quantum Physics}, + year = 2020, + month = jun, + eid = {arXiv:2006.02424}, + pages = {arXiv:2006.02424}, +archivePrefix = {arXiv}, + eprint = {2006.02424}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv200602424T}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + +@article{PhysRevE.96.022140, + title = {Unsupervised learning of phase transitions: From principal component analysis to variational autoencoders}, + author = {Wetzel, Sebastian J.}, + journal = {Phys. Rev. E}, + volume = {96}, + issue = {2}, + pages = {022140}, + numpages = {11}, + year = {2017}, + month = {Aug}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevE.96.022140}, + url = {https://link.aps.org/doi/10.1103/PhysRevE.96.022140} +} + + +@ARTICLE{morawetz2020, + author = {{Morawetz}, Stewart and {De Vlugt}, Isaac J.~S. and {Carrasquilla}, Juan and {Melko}, Roger G.}, + title = "{U(1) symmetric recurrent neural networks for quantum state reconstruction}", + journal = {arXiv e-prints}, + keywords = {Quantum Physics, Condensed Matter - Strongly Correlated Electrons}, + year = 2020, + month = oct, + eid = {arXiv:2010.14514}, + pages = {arXiv:2010.14514}, +archivePrefix = {arXiv}, + eprint = {2010.14514}, + primaryClass = {quant-ph}, + adsurl = {https://ui.adsabs.harvard.edu/abs/2020arXiv201014514M}, + adsnote = {Provided by the SAO/NASA Astrophysics Data System} +} + + + + +@misc{Nori2020, + title={Classification and reconstruction of optical quantum states with deep neural networks}, + author={Shahnawaz Ahmed and Carlos Sánchez Munoz and Franco Nori and Anton Frisk Kockum}, + year={2020}, + eprint={2012.02185}, + archivePrefix={arXiv}, + primaryClass={quant-ph} +} + + +%%%% +%DFT +%%% + +@article{PhysRevLett.108.253002, + title = {Finding Density Functionals with Machine Learning}, + author = {Snyder, John C. and Rupp, Matthias and Hansen, Katja and M\"uller, Klaus-Robert and Burke, Kieron}, + journal = {Phys. Rev. Lett.}, + volume = {108}, + issue = {25}, + pages = {253002}, + numpages = {5}, + year = {2012}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.108.253002}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.108.253002} +} + + + + +@article{doi:10.1063/1.4834075, +author = {Snyder,John C. and Rupp,Matthias and Hansen,Katja and Blooston,Leo and Müller,Klaus-Robert and Burke,Kieron }, +title = {Orbital-free bond breaking via machine learning}, +journal = {The Journal of Chemical Physics}, +volume = {139}, +number = {22}, +pages = {224104}, +year = {2013}, +doi = {10.1063/1.4834075}, + +URL = { + https://doi.org/10.1063/1.4834075 + +}, +eprint = { + https://doi.org/10.1063/1.4834075 + +} + +} + + +@article{silver2016, + title = {Mastering the Game of {{Go}} with Deep Neural Networks and Tree Search}, + author = {Silver, David and Huang, Aja and Maddison, Chris J. and Guez, Arthur and Sifre, Laurent and {van den Driessche}, George and Schrittwieser, Julian and Antonoglou, Ioannis and Panneershelvam, Veda and Lanctot, Marc and Dieleman, Sander and Grewe, Dominik and Nham, John and Kalchbrenner, Nal and Sutskever, Ilya and Lillicrap, Timothy and Leach, Madeleine and Kavukcuoglu, Koray and Graepel, Thore and Hassabis, Demis}, + year = {2016}, + month = jan, + volume = {529}, + pages = {484--489}, + publisher = {{Nature Publishing Group}}, + issn = {1476-4687}, + doi = {10.1038/nature16961}, + abstract = {The game of Go has long been viewed as the most challenging of classic games for artificial intelligence owing to its enormous search space and the difficulty of evaluating board positions and moves. Here we introduce a new approach to computer Go that uses `value networks' to evaluate board positions and `policy networks' to select moves. These deep neural networks are trained by a novel combination of supervised learning from human expert games, and reinforcement learning from games of self-play. Without any lookahead search, the neural networks play Go at the level of state-of-the-art Monte Carlo tree search programs that simulate thousands of random games of self-play. We also introduce a new search algorithm that combines Monte Carlo simulation with value and policy networks. Using this search algorithm, our program AlphaGo achieved a 99.8\% winning rate against other Go programs, and defeated the human European Go champion by 5 games to 0. This is the first time that a computer program has defeated a human professional player in the full-sized game of Go, a feat previously thought to be at least a decade away.}, + journal = {Nature}, + number = {7587} +} + + + + + +@article{PhysRevB.94.245129, + title = {Pure density functional for strong correlation and the thermodynamic limit from machine learning}, + author = {Li, Li and Baker, Thomas E. and White, Steven R. and Burke, Kieron}, + journal = {Phys. Rev. B}, + volume = {94}, + issue = {24}, + pages = {245129}, + numpages = {9}, + year = {2016}, + month = {Dec}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.94.245129}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.94.245129} +} + +@article{rao2018, + title = {Machine Learning the Many-Body Localization Transition in Random Spin Systems}, + author = {Rao, Wen-Jia}, + year = {2018}, + month = sep, + volume = {30}, + pages = {395902}, + publisher = {{IOP Publishing}}, + issn = {0953-8984}, + doi = {10.1088/1361-648X/aaddc6}, + abstract = {The transition between thermal and many-body localized phases in isolated random spin systems are typically identified by the distribution of nearest level spacings. In this work, by employing machine learning methodology, we show this transition can be learnt through raw energy spectrum without any pre-processing. After achieving so in conventional random spin chain with differentiable level spacing distributions, we further construct novel models with misleading signatures of level spacing, and show machine can defeat the latter when training data is raw energy spectrum. Our work shows the low-level energy spectrum contains more information than level spacings, and can be captured by machine in a direct while efficient way, which makes it a promising new tool for studying a variety of isolated quantum systems.}, + journal = {Journal of Physics: Condensed Matter}, + number = {39} +} + +@article{PhysRevLett.120.257204, + title = {Machine Learning Out-of-Equilibrium Phases of Matter}, + author = {Venderley, Jordan and Khemani, Vedika and Kim, Eun-Ah}, + journal = {Phys. Rev. Lett.}, + volume = {120}, + issue = {25}, + pages = {257204}, + numpages = {6}, + year = {2018}, + month = {Jun}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.120.257204}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.120.257204} +} + + + + + +@article{brockherde_bypassing_2017, + title = {Bypassing the {Kohn}-{Sham} equations with machine learning}, + volume = {8}, + issn = {2041-1723}, + url = {https://doi.org/10.1038/s41467-017-00839-3}, + doi = {10.1038/s41467-017-00839-3}, + abstract = {Last year, at least 30,000 scientific papers used the Kohn–Sham scheme of density functional theory to solve electronic structure problems in a wide variety of scientific fields. Machine learning holds the promise of learning the energy functional via examples, bypassing the need to solve the Kohn–Sham equations. This should yield substantial savings in computer time, allowing larger systems and/or longer time-scales to be tackled, but attempts to machine-learn this functional have been limited by the need to find its derivative. The present work overcomes this difficulty by directly learning the density-potential and energy-density maps for test systems and various molecules. We perform the first molecular dynamics simulation with a machine-learned density functional on malonaldehyde and are able to capture the intramolecular proton transfer process. Learning density models now allows the construction of accurate density functionals for realistic molecular systems.}, + number = {1}, + journal = {Nature Communications}, + author = {Brockherde, Felix and Vogt, Leslie and Li, Li and Tuckerman, Mark E. and Burke, Kieron and Müller, Klaus-Robert}, + month = oct, + year = {2017}, + pages = {872}, +} + +@article{Seif_2018, + doi = {10.1088/1361-6455/aad62b}, + url = {https://doi.org/10.1088%2F1361-6455%2Faad62b}, + year = 2018, + month = {aug}, + publisher = {{IOP} Publishing}, + volume = {51}, + number = {17}, + pages = {174006}, + author = {Alireza Seif and Kevin A Landsman and Norbert M Linke and Caroline Figgatt and C Monroe and Mohammad Hafezi}, + title = {Machine learning assisted readout of trapped-ion qubits}, + journal = {Journal of Physics B: Atomic, Molecular and Optical Physics}, + abstract = {We reduce measurement errors in a quantum computer using machine learning techniques. We exploit a simple yet versatile neural network to classify multi-qubit quantum states, which is trained using experimental data. This flexible approach allows the incorporation of any number of features of the data with minimal modifications to the underlying network architecture. We experimentally illustrate this approach in the readout of trapped-ion qubits using additional spatial and temporal features in the data. Using this neural network classifier, we efficiently treat qubit readout crosstalk, resulting in a 30 improvement in detection error over the conventional threshold method. Our approach does not depend on the specific details of the system and can be readily generalized to other quantum computing platforms.} +} + + +@article{PhysRevB.99.075113, + title = {Super-resolving the Ising model with convolutional neural networks}, + author = {Efthymiou, Stavros and Beach, Matthew J. S. and Melko, Roger G.}, + journal = {Phys. Rev. B}, + volume = {99}, + issue = {7}, + pages = {075113}, + numpages = {9}, + year = {2019}, + month = {Feb}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.99.075113}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.99.075113} +} + +@article{PhysRevA.100.022512, + title = {Deep learning and density-functional theory}, + author = {Ryczko, Kevin and Strubbe, David A. and Tamblyn, Isaac}, + journal = {Phys. Rev. A}, + volume = {100}, + issue = {2}, + pages = {022512}, + numpages = {11}, + year = {2019}, + month = {Aug}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.100.022512}, + url = {https://link.aps.org/doi/10.1103/PhysRevA.100.022512} +} + +@article{PhysRevLett.125.076402, + title = {Deep Learning the Hohenberg-Kohn Maps of Density Functional Theory}, + author = {Moreno, Javier Robledo and Carleo, Giuseppe and Georges, Antoine}, + journal = {Phys. Rev. Lett.}, + volume = {125}, + issue = {7}, + pages = {076402}, + numpages = {6}, + year = {2020}, + month = {Aug}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.125.076402}, + url = {https://link.aps.org/doi/10.1103/PhysRevLett.125.076402} +} + +@article{PhysRevResearch.2.033388, + title = {Efficient learning of a one-dimensional density functional theory}, + author = {Denner, M. Michael and Fischer, Mark H. and Neupert, Titus}, + journal = {Phys. Rev. Research}, + volume = {2}, + issue = {3}, + pages = {033388}, + numpages = {9}, + year = {2020}, + month = {Sep}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevResearch.2.033388}, + url = {https://link.aps.org/doi/10.1103/PhysRevResearch.2.033388} +} diff --git a/doc/Articles/fig_qst_beryllium.pdf b/doc/Articles/fig_qst_beryllium.pdf new file mode 100644 index 000000000..fa7db74c8 Binary files /dev/null and b/doc/Articles/fig_qst_beryllium.pdf differ diff --git a/doc/Articles/fig_qst_rydberg.pdf b/doc/Articles/fig_qst_rydberg.pdf new file mode 100644 index 000000000..688474f7e Binary files /dev/null and b/doc/Articles/fig_qst_rydberg.pdf differ diff --git a/doc/Articles/fig_supervised.pdf b/doc/Articles/fig_supervised.pdf new file mode 100644 index 000000000..423f01799 Binary files /dev/null and b/doc/Articles/fig_supervised.pdf differ diff --git a/doc/Articles/fig_vmc.pdf b/doc/Articles/fig_vmc.pdf new file mode 100644 index 000000000..ba93cbe7c Binary files /dev/null and b/doc/Articles/fig_vmc.pdf differ diff --git a/doc/Articles/main.bbl b/doc/Articles/main.bbl new file mode 100644 index 000000000..8bc60bbef --- /dev/null +++ b/doc/Articles/main.bbl @@ -0,0 +1,2688 @@ +%merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked +%Control: key (0) +%Control: author (0) dotless jnrlst +%Control: editor formatted (1) identically to author +%Control: production of article title (0) allowed +%Control: page (1) range +%Control: year (0) verbatim +%Control: production of eprint (0) enabled +\begin{thebibliography}{189}% +\makeatletter +\providecommand \@ifxundefined [1]{% + \@ifx{#1\undefined} +}% +\providecommand \@ifnum [1]{% + \ifnum #1\expandafter \@firstoftwo + \else \expandafter \@secondoftwo + \fi +}% +\providecommand \@ifx [1]{% + \ifx #1\expandafter \@firstoftwo + \else \expandafter \@secondoftwo + \fi +}% +\providecommand \natexlab [1]{#1}% +\providecommand \enquote [1]{``#1''}% +\providecommand \bibnamefont [1]{#1}% +\providecommand \bibfnamefont [1]{#1}% +\providecommand \citenamefont [1]{#1}% +\providecommand \href@noop [0]{\@secondoftwo}% +\providecommand \href [0]{\begingroup \@sanitize@url \@href}% +\providecommand \@href[1]{\@@startlink{#1}\@@href}% +\providecommand \@@href[1]{\endgroup#1\@@endlink}% +\providecommand \@sanitize@url [0]{\catcode `\\12\catcode `\$12\catcode + `\&12\catcode `\#12\catcode `\^12\catcode `\_12\catcode `\%12\relax}% +\providecommand \@@startlink[1]{}% +\providecommand \@@endlink[0]{}% +\providecommand \url [0]{\begingroup\@sanitize@url \@url }% +\providecommand \@url [1]{\endgroup\@href {#1}{\urlprefix }}% +\providecommand \urlprefix [0]{URL }% +\providecommand \Eprint [0]{\href }% +\providecommand \doibase [0]{http://dx.doi.org/}% +\providecommand \selectlanguage [0]{\@gobble}% +\providecommand \bibinfo [0]{\@secondoftwo}% +\providecommand \bibfield [0]{\@secondoftwo}% +\providecommand \translation [1]{[#1]}% +\providecommand \BibitemOpen [0]{}% +\providecommand \bibitemStop [0]{}% +\providecommand \bibitemNoStop [0]{.\EOS\space}% +\providecommand \EOS [0]{\spacefactor3000\relax}% +\providecommand \BibitemShut [1]{\csname bibitem#1\endcsname}% +\let\auto@bib@innerbib\@empty +% +\bibitem [{\citenamefont {Gang}(2007)}]{Xiao:803748}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Wen~Xiao}\ + \bibnamefont {Gang}},\ }\href {\doibase + 10.1093/acprof:oso/9780199227259.001.0001} {\emph {\bibinfo {title} {{Quantum + field theory of many-body systems: from the origin of sound to an origin of + light and electrons}}}}\ (\bibinfo {publisher} {Oxford University Press},\ + \bibinfo {address} {Oxford},\ \bibinfo {year} {2007})\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Carleo}\ \emph {et~al.}(2019)\citenamefont {Carleo}, + \citenamefont {Cirac}, \citenamefont {Cranmer}, \citenamefont {Daudet}, + \citenamefont {Schuld}, \citenamefont {Tishby}, \citenamefont + {Vogt-Maranto},\ and\ \citenamefont {Zdeborov\'a}}]{RevModPhys.91.045002}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Giuseppe}\ + \bibnamefont {Carleo}}, \bibinfo {author} {\bibfnamefont {Ignacio}\ + \bibnamefont {Cirac}}, \bibinfo {author} {\bibfnamefont {Kyle}\ \bibnamefont + {Cranmer}}, \bibinfo {author} {\bibfnamefont {Laurent}\ \bibnamefont + {Daudet}}, \bibinfo {author} {\bibfnamefont {Maria}\ \bibnamefont {Schuld}}, + \bibinfo {author} {\bibfnamefont {Naftali}\ \bibnamefont {Tishby}}, \bibinfo + {author} {\bibfnamefont {Leslie}\ \bibnamefont {Vogt-Maranto}}, \ and\ + \bibinfo {author} {\bibfnamefont {Lenka}\ \bibnamefont {Zdeborov\'a}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Machine learning and the + physical sciences},}\ }\href {\doibase 10.1103/RevModPhys.91.045002} + {\bibfield {journal} {\bibinfo {journal} {Rev. Mod. Phys.}\ }\textbf + {\bibinfo {volume} {91}},\ \bibinfo {pages} {045002} (\bibinfo {year} + {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont + {Carrasquilla}(2020)}]{doi:10.1080/23746149.2020.1797528}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Juan}\ \bibnamefont + {Carrasquilla}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Machine + learning for quantum matter},}\ }\href {\doibase + 10.1080/23746149.2020.1797528} {\bibfield {journal} {\bibinfo {journal} + {Advances in Physics: X}\ }\textbf {\bibinfo {volume} {5}},\ \bibinfo {pages} + {1797528} (\bibinfo {year} {2020})},\ \Eprint + {http://arxiv.org/abs/https://doi.org/10.1080/23746149.2020.1797528} + {https://doi.org/10.1080/23746149.2020.1797528} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Torlai}\ and\ \citenamefont + {Melko}(2020)}]{annurev-conmatphys-031119-050651}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Giacomo}\ \bibnamefont + {Torlai}}\ and\ \bibinfo {author} {\bibfnamefont {Roger~G.}\ \bibnamefont + {Melko}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Machine-learning + quantum states in the nisq era},}\ }\href {\doibase + 10.1146/annurev-conmatphys-031119-050651} {\bibfield {journal} {\bibinfo + {journal} {Annual Review of Condensed Matter Physics}\ }\textbf {\bibinfo + {volume} {11}},\ \bibinfo {pages} {325--344} (\bibinfo {year} {2020})},\ + \Eprint + {http://arxiv.org/abs/https://doi.org/10.1146/annurev-conmatphys-031119-050651} + {https://doi.org/10.1146/annurev-conmatphys-031119-050651} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Carrasquilla}\ and\ \citenamefont + {Melko}(2017)}]{carrasquilla2017nature}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Juan}\ \bibnamefont + {Carrasquilla}}\ and\ \bibinfo {author} {\bibfnamefont {Roger~G.}\ + \bibnamefont {Melko}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Machine learning phases of matter},}\ }\href + {https://doi.org/10.1038/nphys4035} {\bibfield {journal} {\bibinfo + {journal} {Nature Physics}\ }\textbf {\bibinfo {volume} {13}},\ \bibinfo + {pages} {431} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {van Nieuwenburg}\ \emph {et~al.}(2017)\citenamefont + {van Nieuwenburg}, \citenamefont {Liu},\ and\ \citenamefont + {Huber}}]{evert2017nature}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Evert P.~L.}\ + \bibnamefont {van Nieuwenburg}}, \bibinfo {author} {\bibfnamefont {Ye-Hua}\ + \bibnamefont {Liu}}, \ and\ \bibinfo {author} {\bibfnamefont {Sebastian~D.}\ + \bibnamefont {Huber}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Learning phase transitions by confusion},}\ }\href + {https://doi.org/10.1038/nphys4037} {\bibfield {journal} {\bibinfo + {journal} {Nature Physics}\ }\textbf {\bibinfo {volume} {13}},\ \bibinfo + {pages} {435} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Torlai}\ and\ \citenamefont + {Melko}(2016)}]{torlai_learning_2016}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Giacomo}\ \bibnamefont + {Torlai}}\ and\ \bibinfo {author} {\bibfnamefont {Roger~G.}\ \bibnamefont + {Melko}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Learning + thermodynamics with {Boltzmann} machines},}\ }\href {\doibase + 10.1103/PhysRevB.94.165134} {\bibfield {journal} {\bibinfo {journal} + {Physical Review B}\ }\textbf {\bibinfo {volume} {94}},\ \bibinfo {pages} + {165134} (\bibinfo {year} {2016})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Wang}(2016)}]{leiwang2016}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Lei}\ \bibnamefont + {Wang}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Discovering phase + transitions with unsupervised learning},}\ }\href {\doibase + 10.1103/PhysRevB.94.195105} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. B}\ }\textbf {\bibinfo {volume} {94}},\ \bibinfo {pages} {195105} + (\bibinfo {year} {2016})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Ch'ng}\ \emph {et~al.}(2017)\citenamefont {Ch'ng}, + \citenamefont {Carrasquilla}, \citenamefont {Melko},\ and\ \citenamefont + {Khatami}}]{chng2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Kelvin}\ \bibnamefont + {Ch'ng}}, \bibinfo {author} {\bibfnamefont {Juan}\ \bibnamefont + {Carrasquilla}}, \bibinfo {author} {\bibfnamefont {Roger~G.}\ \bibnamefont + {Melko}}, \ and\ \bibinfo {author} {\bibfnamefont {Ehsan}\ \bibnamefont + {Khatami}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Machine + learning phases of strongly correlated fermions},}\ }\href {\doibase + 10.1103/PhysRevX.7.031038} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. X}\ }\textbf {\bibinfo {volume} {7}},\ \bibinfo {pages} {031038} + (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Broecker}\ \emph + {et~al.}(2017{\natexlab{a}})\citenamefont {Broecker}, \citenamefont + {Carrasquilla}, \citenamefont {Melko},\ and\ \citenamefont + {Trebst}}]{broecker2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Peter}\ \bibnamefont + {Broecker}}, \bibinfo {author} {\bibfnamefont {Juan}\ \bibnamefont + {Carrasquilla}}, \bibinfo {author} {\bibfnamefont {Roger~G.}\ \bibnamefont + {Melko}}, \ and\ \bibinfo {author} {\bibfnamefont {Simon}\ \bibnamefont + {Trebst}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Machine learning + quantum phases of matter beyond the fermion sign problem},}\ }\href {\doibase + 10.1038/s41598-017-09098-0} {\bibfield {journal} {\bibinfo {journal} + {Scientific Reports}\ }\textbf {\bibinfo {volume} {7}},\ \bibinfo {pages} + {8823} (\bibinfo {year} {2017}{\natexlab{a}})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Zhang}\ and\ \citenamefont {Kim}(2017)}]{eun-ah2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Yi}~\bibnamefont + {Zhang}}\ and\ \bibinfo {author} {\bibfnamefont {Eun-Ah}\ \bibnamefont + {Kim}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum loop + topography for machine learning},}\ }\href {\doibase + 10.1103/PhysRevLett.118.216401} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {118}},\ \bibinfo {pages} + {216401} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Deng}\ \emph {et~al.}(2017)\citenamefont {Deng}, + \citenamefont {Li},\ and\ \citenamefont {Das~Sarma}}]{dassarma2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Dong-Ling}\ + \bibnamefont {Deng}}, \bibinfo {author} {\bibfnamefont {Xiaopeng}\ + \bibnamefont {Li}}, \ and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont + {Das~Sarma}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Machine + learning topological states},}\ }\href {\doibase 10.1103/PhysRevB.96.195145} + {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo + {volume} {96}},\ \bibinfo {pages} {195145} (\bibinfo {year} + {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Schindler}\ \emph {et~al.}(2017)\citenamefont + {Schindler}, \citenamefont {Regnault},\ and\ \citenamefont + {Neupert}}]{neupeurt2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Frank}\ \bibnamefont + {Schindler}}, \bibinfo {author} {\bibfnamefont {Nicolas}\ \bibnamefont + {Regnault}}, \ and\ \bibinfo {author} {\bibfnamefont {Titus}\ \bibnamefont + {Neupert}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Probing + many-body localization with neural networks},}\ }\href {\doibase + 10.1103/PhysRevB.95.245134} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. B}\ }\textbf {\bibinfo {volume} {95}},\ \bibinfo {pages} {245134} + (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Hsu}\ \emph {et~al.}(2018)\citenamefont {Hsu}, + \citenamefont {Li}, \citenamefont {Deng},\ and\ \citenamefont + {Das~Sarma}}]{yi-ting2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Yi-Ting}\ \bibnamefont + {Hsu}}, \bibinfo {author} {\bibfnamefont {Xiao}\ \bibnamefont {Li}}, \bibinfo + {author} {\bibfnamefont {Dong-Ling}\ \bibnamefont {Deng}}, \ and\ \bibinfo + {author} {\bibfnamefont {S.}~\bibnamefont {Das~Sarma}},\ }\bibfield {title} + {\enquote {\bibinfo {title} {Machine learning many-body localization: Search + for the elusive nonergodic metal},}\ }\href {\doibase + 10.1103/PhysRevLett.121.245701} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {121}},\ \bibinfo {pages} + {245701} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Huembeli}\ \emph {et~al.}(2018)\citenamefont + {Huembeli}, \citenamefont {Dauphin},\ and\ \citenamefont + {Wittek}}]{huembeli2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Patrick}\ \bibnamefont + {Huembeli}}, \bibinfo {author} {\bibfnamefont {Alexandre}\ \bibnamefont + {Dauphin}}, \ and\ \bibinfo {author} {\bibfnamefont {Peter}\ \bibnamefont + {Wittek}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Identifying + quantum phase transitions with adversarial neural networks},}\ }\href + {\doibase 10.1103/PhysRevB.97.134109} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {97}},\ \bibinfo + {pages} {134109} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Greitemann}\ \emph {et~al.}(2019)\citenamefont + {Greitemann}, \citenamefont {Liu},\ and\ \citenamefont + {Pollet}}]{PhysRevB.99.060404}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Jonas}\ \bibnamefont + {Greitemann}}, \bibinfo {author} {\bibfnamefont {Ke}~\bibnamefont {Liu}}, \ + and\ \bibinfo {author} {\bibfnamefont {Lode}\ \bibnamefont {Pollet}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Probing hidden spin order + with interpretable machine learning},}\ }\href {\doibase + 10.1103/PhysRevB.99.060404} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. B}\ }\textbf {\bibinfo {volume} {99}},\ \bibinfo {pages} {060404} + (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Liu}\ \emph {et~al.}(2019)\citenamefont {Liu}, + \citenamefont {Greitemann},\ and\ \citenamefont + {Pollet}}]{PhysRevB.99.104410}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Ke}~\bibnamefont + {Liu}}, \bibinfo {author} {\bibfnamefont {Jonas}\ \bibnamefont {Greitemann}}, + \ and\ \bibinfo {author} {\bibfnamefont {Lode}\ \bibnamefont {Pollet}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Learning multiple order + parameters with interpretable machines},}\ }\href {\doibase + 10.1103/PhysRevB.99.104410} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. B}\ }\textbf {\bibinfo {volume} {99}},\ \bibinfo {pages} {104410} + (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Zhang}\ \emph {et~al.}(2019)\citenamefont {Zhang}, + \citenamefont {Mesaros}, \citenamefont {Fujita}, \citenamefont {Edkins}, + \citenamefont {Hamidian}, \citenamefont {Ch'ng}, \citenamefont {Eisaki}, + \citenamefont {Uchida}, \citenamefont {Davis}, \citenamefont {Khatami},\ and\ + \citenamefont {Kim}}]{Zhang_MLcuprates}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Yi}~\bibnamefont + {Zhang}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Mesaros}}, + \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Fujita}}, \bibinfo + {author} {\bibfnamefont {S.~D.}\ \bibnamefont {Edkins}}, \bibinfo {author} + {\bibfnamefont {M.~H.}\ \bibnamefont {Hamidian}}, \bibinfo {author} + {\bibfnamefont {K.}~\bibnamefont {Ch'ng}}, \bibinfo {author} {\bibfnamefont + {H.}~\bibnamefont {Eisaki}}, \bibinfo {author} {\bibfnamefont + {S.}~\bibnamefont {Uchida}}, \bibinfo {author} {\bibfnamefont + {J.~C.~S{\'e}amus}\ \bibnamefont {Davis}}, \bibinfo {author} {\bibfnamefont + {Ehsan}\ \bibnamefont {Khatami}}, \ and\ \bibinfo {author} {\bibfnamefont + {Eun-Ah}\ \bibnamefont {Kim}},\ }\bibfield {title} {\enquote {\bibinfo + {title} {Machine learning in electronic-quantum-matter imaging + experiments},}\ }\href {\doibase 10.1038/s41586-019-1319-8} {\bibfield + {journal} {\bibinfo {journal} {Nature}\ }\textbf {\bibinfo {volume} {570}},\ + \bibinfo {pages} {484--490} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Bohrdt}\ \emph {et~al.}(2018)\citenamefont {Bohrdt}, + \citenamefont {Chiu}, \citenamefont {Ji}, \citenamefont {Xu}, \citenamefont + {Greif}, \citenamefont {Greiner}, \citenamefont {Demler}, \citenamefont + {Grusdt},\ and\ \citenamefont {Knap}}]{Bohrdt2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Annabelle}\ + \bibnamefont {Bohrdt}}, \bibinfo {author} {\bibfnamefont {Christie~S.}\ + \bibnamefont {Chiu}}, \bibinfo {author} {\bibfnamefont {Geoffrey}\ + \bibnamefont {Ji}}, \bibinfo {author} {\bibfnamefont {Muqing}\ \bibnamefont + {Xu}}, \bibinfo {author} {\bibfnamefont {Daniel}\ \bibnamefont {Greif}}, + \bibinfo {author} {\bibfnamefont {Markus}\ \bibnamefont {Greiner}}, \bibinfo + {author} {\bibfnamefont {Eugene}\ \bibnamefont {Demler}}, \bibinfo {author} + {\bibfnamefont {Fabian}\ \bibnamefont {Grusdt}}, \ and\ \bibinfo {author} + {\bibfnamefont {Michael}\ \bibnamefont {Knap}},\ }\bibfield {title} + {\enquote {\bibinfo {title} {{Classifying Snapshots of the Doped Hubbard + Model with Machine Learning}},}\ }\href {http://arxiv.org/abs/1811.12425} {\ + (\bibinfo {year} {2018})},\ \Eprint {http://arxiv.org/abs/1811.12425} + {arXiv:1811.12425} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Rem}\ \emph {et~al.}(2018)\citenamefont {Rem}, + \citenamefont {K{\"{a}}ming}, \citenamefont {Tarnowski}, \citenamefont + {Asteria}, \citenamefont {Fl{\"{a}}schner}, \citenamefont {Becker}, + \citenamefont {Sengstock},\ and\ \citenamefont {Weitenberg}}]{Rem2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Benno~S.}\ + \bibnamefont {Rem}}, \bibinfo {author} {\bibfnamefont {Niklas}\ \bibnamefont + {K{\"{a}}ming}}, \bibinfo {author} {\bibfnamefont {Matthias}\ \bibnamefont + {Tarnowski}}, \bibinfo {author} {\bibfnamefont {Luca}\ \bibnamefont + {Asteria}}, \bibinfo {author} {\bibfnamefont {Nick}\ \bibnamefont + {Fl{\"{a}}schner}}, \bibinfo {author} {\bibfnamefont {Christoph}\ + \bibnamefont {Becker}}, \bibinfo {author} {\bibfnamefont {Klaus}\ + \bibnamefont {Sengstock}}, \ and\ \bibinfo {author} {\bibfnamefont + {Christof}\ \bibnamefont {Weitenberg}},\ }\bibfield {title} {\enquote + {\bibinfo {title} {{Identifying Quantum Phase Transitions using Artificial + Neural Networks on Experimental Data}},}\ }\href + {http://arxiv.org/abs/1809.05519} {\ (\bibinfo {year} {2018})},\ \Eprint + {http://arxiv.org/abs/1809.05519} {arXiv:1809.05519} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Lian}\ \emph {et~al.}(2019)\citenamefont {Lian}, + \citenamefont {Wang}, \citenamefont {Lu}, \citenamefont {Huang}, + \citenamefont {Wang}, \citenamefont {Yuan}, \citenamefont {Zhang}, + \citenamefont {Ouyang}, \citenamefont {Wang}, \citenamefont {Huang}, + \citenamefont {He}, \citenamefont {Chang}, \citenamefont {Deng},\ and\ + \citenamefont {Duan}}]{PhysRevLett.122.210503}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Wenqian}\ \bibnamefont + {Lian}}, \bibinfo {author} {\bibfnamefont {Sheng-Tao}\ \bibnamefont {Wang}}, + \bibinfo {author} {\bibfnamefont {Sirui}\ \bibnamefont {Lu}}, \bibinfo + {author} {\bibfnamefont {Yuanyuan}\ \bibnamefont {Huang}}, \bibinfo {author} + {\bibfnamefont {Fei}\ \bibnamefont {Wang}}, \bibinfo {author} {\bibfnamefont + {Xinxing}\ \bibnamefont {Yuan}}, \bibinfo {author} {\bibfnamefont {Wengang}\ + \bibnamefont {Zhang}}, \bibinfo {author} {\bibfnamefont {Xiaolong}\ + \bibnamefont {Ouyang}}, \bibinfo {author} {\bibfnamefont {Xin}\ \bibnamefont + {Wang}}, \bibinfo {author} {\bibfnamefont {Xianzhi}\ \bibnamefont {Huang}}, + \bibinfo {author} {\bibfnamefont {Li}~\bibnamefont {He}}, \bibinfo {author} + {\bibfnamefont {Xiuying}\ \bibnamefont {Chang}}, \bibinfo {author} + {\bibfnamefont {Dong-Ling}\ \bibnamefont {Deng}}, \ and\ \bibinfo {author} + {\bibfnamefont {Luming}\ \bibnamefont {Duan}},\ }\bibfield {title} {\enquote + {\bibinfo {title} {Machine learning topological phases with a solid-state + quantum simulator},}\ }\href {\doibase 10.1103/PhysRevLett.122.210503} + {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf + {\bibinfo {volume} {122}},\ \bibinfo {pages} {210503} (\bibinfo {year} + {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Huang}\ and\ \citenamefont {Wang}(2017)}]{huang2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Li}~\bibnamefont + {Huang}}\ and\ \bibinfo {author} {\bibfnamefont {Lei}\ \bibnamefont {Wang}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Accelerated monte carlo + simulations with restricted boltzmann machines},}\ }\href {\doibase + 10.1103/PhysRevB.95.035105} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. B}\ }\textbf {\bibinfo {volume} {95}},\ \bibinfo {pages} {035105} + (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Liu}\ \emph {et~al.}(2017)\citenamefont {Liu}, + \citenamefont {Qi}, \citenamefont {Meng},\ and\ \citenamefont + {Fu}}]{junwei2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Junwei}\ \bibnamefont + {Liu}}, \bibinfo {author} {\bibfnamefont {Yang}\ \bibnamefont {Qi}}, \bibinfo + {author} {\bibfnamefont {Zi~Yang}\ \bibnamefont {Meng}}, \ and\ \bibinfo + {author} {\bibfnamefont {Liang}\ \bibnamefont {Fu}},\ }\bibfield {title} + {\enquote {\bibinfo {title} {Self-learning monte carlo method},}\ }\href + {\doibase 10.1103/PhysRevB.95.041101} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {95}},\ \bibinfo + {pages} {041101} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Xu}\ \emph {et~al.}(2017)\citenamefont {Xu}, + \citenamefont {Qi}, \citenamefont {Liu}, \citenamefont {Fu},\ and\ + \citenamefont {Meng}}]{xiao_yan2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Xiao~Yan}\ + \bibnamefont {Xu}}, \bibinfo {author} {\bibfnamefont {Yang}\ \bibnamefont + {Qi}}, \bibinfo {author} {\bibfnamefont {Junwei}\ \bibnamefont {Liu}}, + \bibinfo {author} {\bibfnamefont {Liang}\ \bibnamefont {Fu}}, \ and\ \bibinfo + {author} {\bibfnamefont {Zi~Yang}\ \bibnamefont {Meng}},\ }\bibfield {title} + {\enquote {\bibinfo {title} {Self-learning quantum monte carlo method in + interacting fermion systems},}\ }\href {\doibase 10.1103/PhysRevB.96.041119} + {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo + {volume} {96}},\ \bibinfo {pages} {041119} (\bibinfo {year} + {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Inack}\ \emph {et~al.}(2018)\citenamefont {Inack}, + \citenamefont {Santoro}, \citenamefont {Dell'Anna},\ and\ \citenamefont + {Pilati}}]{inack2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~M.}\ \bibnamefont + {Inack}}, \bibinfo {author} {\bibfnamefont {G.~E.}\ \bibnamefont {Santoro}}, + \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Dell'Anna}}, \ and\ + \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Pilati}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Projective quantum monte carlo + simulations guided by unrestricted neural network states},}\ }\href {\doibase + 10.1103/PhysRevB.98.235145} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. B}\ }\textbf {\bibinfo {volume} {98}},\ \bibinfo {pages} {235145} + (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Parolini}\ \emph {et~al.}(2019)\citenamefont + {Parolini}, \citenamefont {Inack}, \citenamefont {Giudici},\ and\ + \citenamefont {Pilati}}]{parolini2019}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont + {Parolini}}, \bibinfo {author} {\bibfnamefont {E.~M.}\ \bibnamefont {Inack}}, + \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Giudici}}, \ and\ + \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Pilati}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Tunneling in projective quantum monte + carlo simulations with guiding wave functions},}\ }\href {\doibase + 10.1103/PhysRevB.100.214303} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {100}},\ \bibinfo {pages} + {214303} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Pilati}\ \emph {et~al.}(2019)\citenamefont {Pilati}, + \citenamefont {Inack},\ and\ \citenamefont {Pieri}}]{pilati2019}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont + {Pilati}}, \bibinfo {author} {\bibfnamefont {E.~M.}\ \bibnamefont {Inack}}, \ + and\ \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Pieri}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Self-learning projective quantum monte + carlo simulations guided by restricted boltzmann machines},}\ }\href + {\doibase 10.1103/PhysRevE.100.043301} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. E}\ }\textbf {\bibinfo {volume} {100}},\ \bibinfo + {pages} {043301} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {McNaughton}\ \emph {et~al.}(2020)\citenamefont + {McNaughton}, \citenamefont {Milo\ifmmode \check{s}\else + \v{s}\fi{}evi\ifmmode~\acute{c}\else \'{c}\fi{}}, \citenamefont {Perali},\ + and\ \citenamefont {Pilati}}]{mcnaughton2020}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont + {McNaughton}}, \bibinfo {author} {\bibfnamefont {M.~V.}\ \bibnamefont + {Milo\ifmmode \check{s}\else \v{s}\fi{}evi\ifmmode~\acute{c}\else + \'{c}\fi{}}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Perali}}, \ + and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Pilati}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Boosting monte carlo + simulations of spin glasses using autoregressive neural networks},}\ }\href + {\doibase 10.1103/PhysRevE.101.053312} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. E}\ }\textbf {\bibinfo {volume} {101}},\ \bibinfo + {pages} {053312} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Albergo}\ \emph {et~al.}(2019)\citenamefont + {Albergo}, \citenamefont {Kanwar},\ and\ \citenamefont + {Shanahan}}]{albergo2019}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~S.}\ \bibnamefont + {Albergo}}, \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Kanwar}}, \ + and\ \bibinfo {author} {\bibfnamefont {P.~E.}\ \bibnamefont {Shanahan}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Flow-based generative models + for markov chain monte carlo in lattice field theory},}\ }\href {\doibase + 10.1103/PhysRevD.100.034515} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. D}\ }\textbf {\bibinfo {volume} {100}},\ \bibinfo {pages} + {034515} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Wu}\ \emph {et~al.}(2019)\citenamefont {Wu}, + \citenamefont {Wang},\ and\ \citenamefont {Zhang}}]{Wu_2019}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Dian}\ \bibnamefont + {Wu}}, \bibinfo {author} {\bibfnamefont {Lei}\ \bibnamefont {Wang}}, \ and\ + \bibinfo {author} {\bibfnamefont {Pan}\ \bibnamefont {Zhang}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Solving statistical mechanics using + variational autoregressive networks},}\ }\href {\doibase + 10.1103/PhysRevLett.122.080602} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {122}},\ \bibinfo {pages} + {080602} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Androsiuk}\ \emph {et~al.}(1993)\citenamefont + {Androsiuk}, \citenamefont {Kulak},\ and\ \citenamefont + {Sienicki}}]{androsiuk1993}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont + {Androsiuk}}, \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Kulak}}, \ + and\ \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Sienicki}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Neural network solution of + the schroedinger equation for a two-dimensional harmonic oscillator},}\ + }\href {\doibase https://doi.org/10.1016/0301-0104(93)80153-Z} {\bibfield + {journal} {\bibinfo {journal} {Chemical Physics}\ }\textbf {\bibinfo + {volume} {173}},\ \bibinfo {pages} {377 -- 383} (\bibinfo {year} + {1993})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Lagaris}}\ \emph {et~al.}(1998)\citenamefont + {{Lagaris}}, \citenamefont {{Likas}},\ and\ \citenamefont + {{Fotiadis}}}]{LAGARIS19971}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {I.~E.}\ \bibnamefont + {{Lagaris}}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {{Likas}}}, + \ and\ \bibinfo {author} {\bibfnamefont {D.~I.}\ \bibnamefont {{Fotiadis}}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Artificial neural networks + for solving ordinary and partial differential equations},}\ }\href {\doibase + 10.1109/72.712178} {\bibfield {journal} {\bibinfo {journal} {IEEE + Transactions on Neural Networks}\ }\textbf {\bibinfo {volume} {9}},\ \bibinfo + {pages} {987--1000} (\bibinfo {year} {1998})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Carleo}\ and\ \citenamefont + {Troyer}(2017)}]{Carleo_2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Giuseppe}\ + \bibnamefont {Carleo}}\ and\ \bibinfo {author} {\bibfnamefont {Matthias}\ + \bibnamefont {Troyer}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Solving the quantum many-body problem with artificial neural networks},}\ + }\href {\doibase 10.1126/science.aag2302} {\bibfield {journal} {\bibinfo + {journal} {Science}\ }\textbf {\bibinfo {volume} {355}},\ \bibinfo {pages} + {602--606} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Cai}\ and\ \citenamefont {Liu}(2018)}]{zi2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Zi}~\bibnamefont + {Cai}}\ and\ \bibinfo {author} {\bibfnamefont {Jinguo}\ \bibnamefont {Liu}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Approximating quantum + many-body wave functions using artificial neural networks},}\ }\href + {\doibase 10.1103/PhysRevB.97.035116} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {97}},\ \bibinfo + {pages} {035116} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Luo}\ and\ \citenamefont {Clark}(2019)}]{Di_Luo}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Di}~\bibnamefont + {Luo}}\ and\ \bibinfo {author} {\bibfnamefont {Bryan~K.}\ \bibnamefont + {Clark}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Backflow + transformations via neural networks for quantum many-body wave functions},}\ + }\href {\doibase 10.1103/PhysRevLett.122.226401} {\bibfield {journal} + {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {122}},\ + \bibinfo {pages} {226401} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Pfau}\ \emph {et~al.}(2020)\citenamefont {Pfau}, + \citenamefont {Spencer}, \citenamefont {Matthews},\ and\ \citenamefont + {Foulkes}}]{pfau2019abinitio}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {David}\ \bibnamefont + {Pfau}}, \bibinfo {author} {\bibfnamefont {James~S.}\ \bibnamefont + {Spencer}}, \bibinfo {author} {\bibfnamefont {Alexander G. D.~G.}\ + \bibnamefont {Matthews}}, \ and\ \bibinfo {author} {\bibfnamefont {W.~M.~C.}\ + \bibnamefont {Foulkes}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Ab + initio solution of the many-electron schr\"odinger equation with deep neural + networks},}\ }\href {\doibase 10.1103/PhysRevResearch.2.033429} {\bibfield + {journal} {\bibinfo {journal} {Phys. Rev. Research}\ }\textbf {\bibinfo + {volume} {2}},\ \bibinfo {pages} {033429} (\bibinfo {year} + {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Hermann}\ \emph {et~al.}(2020)\citenamefont + {Hermann}, \citenamefont {Schätzle},\ and\ \citenamefont + {Noé}}]{hermann2019deep}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Jan}\ \bibnamefont + {Hermann}}, \bibinfo {author} {\bibfnamefont {Zeno}\ \bibnamefont + {Schätzle}}, \ and\ \bibinfo {author} {\bibfnamefont {Frank}\ \bibnamefont + {Noé}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Deep-neural-network solution of the electronic {Schrödinger} equation},}\ + }\href {\doibase 10.1038/s41557-020-0544-y} {\bibfield {journal} {\bibinfo + {journal} {Nature Chemistry}\ }\textbf {\bibinfo {volume} {12}},\ \bibinfo + {pages} {891--897} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Hartmann}\ and\ \citenamefont + {Carleo}(2019)}]{PhysRevLett.122.250502}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Michael~J.}\ + \bibnamefont {Hartmann}}\ and\ \bibinfo {author} {\bibfnamefont {Giuseppe}\ + \bibnamefont {Carleo}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Neural-network approach to dissipative quantum many-body dynamics},}\ }\href + {\doibase 10.1103/PhysRevLett.122.250502} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {122}},\ \bibinfo + {pages} {250502} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Nagy}\ and\ \citenamefont + {Savona}(2019)}]{PhysRevLett.122.250501}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Alexandra}\ + \bibnamefont {Nagy}}\ and\ \bibinfo {author} {\bibfnamefont {Vincenzo}\ + \bibnamefont {Savona}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Variational quantum monte carlo method with a neural-network ansatz for open + quantum systems},}\ }\href {\doibase 10.1103/PhysRevLett.122.250501} + {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf + {\bibinfo {volume} {122}},\ \bibinfo {pages} {250501} (\bibinfo {year} + {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Vicentini}\ \emph {et~al.}(2019)\citenamefont + {Vicentini}, \citenamefont {Biella}, \citenamefont {Regnault},\ and\ + \citenamefont {Ciuti}}]{PhysRevLett.122.250503}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Filippo}\ \bibnamefont + {Vicentini}}, \bibinfo {author} {\bibfnamefont {Alberto}\ \bibnamefont + {Biella}}, \bibinfo {author} {\bibfnamefont {Nicolas}\ \bibnamefont + {Regnault}}, \ and\ \bibinfo {author} {\bibfnamefont {Cristiano}\ + \bibnamefont {Ciuti}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Variational neural-network ansatz for steady states in open quantum + systems},}\ }\href {\doibase 10.1103/PhysRevLett.122.250503} {\bibfield + {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo + {volume} {122}},\ \bibinfo {pages} {250503} (\bibinfo {year} + {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Yoshioka}\ and\ \citenamefont + {Hamazaki}(2019)}]{PhysRevB.99.214306}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Nobuyuki}\ + \bibnamefont {Yoshioka}}\ and\ \bibinfo {author} {\bibfnamefont {Ryusuke}\ + \bibnamefont {Hamazaki}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Constructing neural stationary states for open quantum many-body systems},}\ + }\href {\doibase 10.1103/PhysRevB.99.214306} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {99}},\ \bibinfo + {pages} {214306} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Choo}\ \emph {et~al.}(2020)\citenamefont {Choo}, + \citenamefont {Mezzacapo},\ and\ \citenamefont + {Carleo}}]{choo_fermionicnqs2020}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Kenny}\ \bibnamefont + {Choo}}, \bibinfo {author} {\bibfnamefont {Antonio}\ \bibnamefont + {Mezzacapo}}, \ and\ \bibinfo {author} {\bibfnamefont {Giuseppe}\ + \bibnamefont {Carleo}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Fermionic neural-network states for ab-initio electronic structure},}\ + }\href {\doibase 10.1038/s41467-020-15724-9} {\bibfield {journal} {\bibinfo + {journal} {Nature Communications}\ }\textbf {\bibinfo {volume} {11}},\ + \bibinfo {pages} {2368} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Sharir}\ \emph {et~al.}(2020)\citenamefont {Sharir}, + \citenamefont {Levine}, \citenamefont {Wies}, \citenamefont {Carleo},\ and\ + \citenamefont {Shashua}}]{PhysRevLett.124.020503}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Or}~\bibnamefont + {Sharir}}, \bibinfo {author} {\bibfnamefont {Yoav}\ \bibnamefont {Levine}}, + \bibinfo {author} {\bibfnamefont {Noam}\ \bibnamefont {Wies}}, \bibinfo + {author} {\bibfnamefont {Giuseppe}\ \bibnamefont {Carleo}}, \ and\ \bibinfo + {author} {\bibfnamefont {Amnon}\ \bibnamefont {Shashua}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Deep autoregressive models for the + efficient variational simulation of many-body quantum systems},}\ }\href + {\doibase 10.1103/PhysRevLett.124.020503} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {124}},\ \bibinfo + {pages} {020503} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Hibat-Allah}\ \emph {et~al.}(2020)\citenamefont + {Hibat-Allah}, \citenamefont {Ganahl}, \citenamefont {Hayward}, \citenamefont + {Melko},\ and\ \citenamefont {Carrasquilla}}]{RNNWF_2020}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Mohamed}\ \bibnamefont + {Hibat-Allah}}, \bibinfo {author} {\bibfnamefont {Martin}\ \bibnamefont + {Ganahl}}, \bibinfo {author} {\bibfnamefont {Lauren~E.}\ \bibnamefont + {Hayward}}, \bibinfo {author} {\bibfnamefont {Roger~G.}\ \bibnamefont + {Melko}}, \ and\ \bibinfo {author} {\bibfnamefont {Juan}\ \bibnamefont + {Carrasquilla}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Recurrent + neural network wave functions},}\ }\href {\doibase + 10.1103/PhysRevResearch.2.023358} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. Research}\ }\textbf {\bibinfo {volume} {2}},\ \bibinfo {pages} + {023358} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Roth}}(2020)}]{roth2020iterative}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Christopher}\ + \bibnamefont {{Roth}}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {{Iterative Retraining of Quantum Spin Models Using Recurrent Neural + Networks}},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} + {arXiv e-prints}\ ,\ \bibinfo {eid} {arXiv:2003.06228}} (\bibinfo {year} + {2020})},\ \Eprint {http://arxiv.org/abs/2003.06228} {arXiv:2003.06228 + [physics.comp-ph]} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Bukov}\ \emph + {et~al.}(2018{\natexlab{a}})\citenamefont {Bukov}, \citenamefont {Day}, + \citenamefont {Sels}, \citenamefont {Weinberg}, \citenamefont {Polkovnikov},\ + and\ \citenamefont {Mehta}}]{PhysRevX.8.031086}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Marin}\ \bibnamefont + {Bukov}}, \bibinfo {author} {\bibfnamefont {Alexandre G.~R.}\ \bibnamefont + {Day}}, \bibinfo {author} {\bibfnamefont {Dries}\ \bibnamefont {Sels}}, + \bibinfo {author} {\bibfnamefont {Phillip}\ \bibnamefont {Weinberg}}, + \bibinfo {author} {\bibfnamefont {Anatoli}\ \bibnamefont {Polkovnikov}}, \ + and\ \bibinfo {author} {\bibfnamefont {Pankaj}\ \bibnamefont {Mehta}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Reinforcement learning in + different phases of quantum control},}\ }\href {\doibase + 10.1103/PhysRevX.8.031086} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. X}\ }\textbf {\bibinfo {volume} {8}},\ \bibinfo {pages} {031086} + (\bibinfo {year} {2018}{\natexlab{a}})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {F\"osel}\ \emph {et~al.}(2018)\citenamefont + {F\"osel}, \citenamefont {Tighineanu}, \citenamefont {Weiss},\ and\ + \citenamefont {Marquardt}}]{PhysRevX.8.031084}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Thomas}\ \bibnamefont + {F\"osel}}, \bibinfo {author} {\bibfnamefont {Petru}\ \bibnamefont + {Tighineanu}}, \bibinfo {author} {\bibfnamefont {Talitha}\ \bibnamefont + {Weiss}}, \ and\ \bibinfo {author} {\bibfnamefont {Florian}\ \bibnamefont + {Marquardt}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Reinforcement + learning with neural networks for quantum feedback},}\ }\href {\doibase + 10.1103/PhysRevX.8.031084} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. X}\ }\textbf {\bibinfo {volume} {8}},\ \bibinfo {pages} {031084} + (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Day}\ \emph {et~al.}(2019)\citenamefont {Day}, + \citenamefont {Bukov}, \citenamefont {Weinberg}, \citenamefont {Mehta},\ and\ + \citenamefont {Sels}}]{PhysRevLett.122.020601}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Alexandre G.~R.}\ + \bibnamefont {Day}}, \bibinfo {author} {\bibfnamefont {Marin}\ \bibnamefont + {Bukov}}, \bibinfo {author} {\bibfnamefont {Phillip}\ \bibnamefont + {Weinberg}}, \bibinfo {author} {\bibfnamefont {Pankaj}\ \bibnamefont + {Mehta}}, \ and\ \bibinfo {author} {\bibfnamefont {Dries}\ \bibnamefont + {Sels}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Glassy phase of + optimal quantum control},}\ }\href {\doibase 10.1103/PhysRevLett.122.020601} + {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf + {\bibinfo {volume} {122}},\ \bibinfo {pages} {020601} (\bibinfo {year} + {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Niu}\ \emph {et~al.}(2019)\citenamefont {Niu}, + \citenamefont {Boixo}, \citenamefont {Smelyanskiy},\ and\ \citenamefont + {Neven}}]{niu_universal_2019}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Murphy~Yuezhen}\ + \bibnamefont {Niu}}, \bibinfo {author} {\bibfnamefont {Sergio}\ \bibnamefont + {Boixo}}, \bibinfo {author} {\bibfnamefont {Vadim~N.}\ \bibnamefont + {Smelyanskiy}}, \ and\ \bibinfo {author} {\bibfnamefont {Hartmut}\ + \bibnamefont {Neven}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Universal quantum control through deep reinforcement learning},}\ }\href + {\doibase 10.1038/s41534-019-0141-3} {\bibfield {journal} {\bibinfo + {journal} {npj Quantum Information}\ }\textbf {\bibinfo {volume} {5}},\ + \bibinfo {pages} {1--8} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Yao}}\ \emph {et~al.}(2020)\citenamefont {{Yao}}, + \citenamefont {{Lin}},\ and\ \citenamefont {{Bukov}}}]{2020arXiv201003655Y}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Jiahao}\ \bibnamefont + {{Yao}}}, \bibinfo {author} {\bibfnamefont {Lin}\ \bibnamefont {{Lin}}}, \ + and\ \bibinfo {author} {\bibfnamefont {Marin}\ \bibnamefont {{Bukov}}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {{Reinforcement Learning for + Many-Body Ground State Preparation based on Counter-Diabatic Driving}},}\ + }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {arXiv e-prints}\ + ,\ \bibinfo {eid} {arXiv:2010.03655}} (\bibinfo {year} {2020})},\ \Eprint + {http://arxiv.org/abs/2010.03655} {arXiv:2010.03655 [quant-ph]} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Coopmans}\ \emph {et~al.}(2020)\citenamefont + {Coopmans}, \citenamefont {Luo}, \citenamefont {Kells}, \citenamefont + {Clark},\ and\ \citenamefont {Carrasquilla}}]{coopmans2020}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Luuk}\ \bibnamefont + {Coopmans}}, \bibinfo {author} {\bibfnamefont {Di}~\bibnamefont {Luo}}, + \bibinfo {author} {\bibfnamefont {Graham}\ \bibnamefont {Kells}}, \bibinfo + {author} {\bibfnamefont {Bryan~K.}\ \bibnamefont {Clark}}, \ and\ \bibinfo + {author} {\bibfnamefont {Juan}\ \bibnamefont {Carrasquilla}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Protocol {{Discovery}} for the {{Quantum + Control}} of {{Majoranas}} by {{Differential Programming}} and {{Natural + Evolution Strategies}}},}\ }\href@noop {} {\bibfield {journal} {\bibinfo + {journal} {arXiv:2008.09128 [cond-mat, physics:physics, physics:quant-ph]}\ } + (\bibinfo {year} {2020})},\ \Eprint {http://arxiv.org/abs/2008.09128} + {arXiv:2008.09128 [cond-mat, physics:physics, physics:quant-ph]} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Torlai}\ \emph {et~al.}(2018)\citenamefont {Torlai}, + \citenamefont {Mazzola}, \citenamefont {Carrasquilla}, \citenamefont + {Troyer}, \citenamefont {Melko},\ and\ \citenamefont {Carleo}}]{torlai_Tomo}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Giacomo}\ \bibnamefont + {Torlai}}, \bibinfo {author} {\bibfnamefont {Guglielmo}\ \bibnamefont + {Mazzola}}, \bibinfo {author} {\bibfnamefont {Juan}\ \bibnamefont + {Carrasquilla}}, \bibinfo {author} {\bibfnamefont {Matthias}\ \bibnamefont + {Troyer}}, \bibinfo {author} {\bibfnamefont {Roger}\ \bibnamefont {Melko}}, \ + and\ \bibinfo {author} {\bibfnamefont {Giuseppe}\ \bibnamefont {Carleo}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Neural-network quantum state + tomography},}\ }\href {\doibase 10.1038/s41567-018-0048-5} {\bibfield + {journal} {\bibinfo {journal} {Nature Physics}\ }\textbf {\bibinfo {volume} + {14}},\ \bibinfo {pages} {447--450} (\bibinfo {year} {2018})}\BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Rocchetto}\ \emph {et~al.}(2018)\citenamefont + {Rocchetto}, \citenamefont {Grant}, \citenamefont {Strelchuk}, \citenamefont + {Carleo},\ and\ \citenamefont {Severini}}]{rocchetto}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Andrea}\ \bibnamefont + {Rocchetto}}, \bibinfo {author} {\bibfnamefont {Edward}\ \bibnamefont + {Grant}}, \bibinfo {author} {\bibfnamefont {Sergii}\ \bibnamefont + {Strelchuk}}, \bibinfo {author} {\bibfnamefont {Giuseppe}\ \bibnamefont + {Carleo}}, \ and\ \bibinfo {author} {\bibfnamefont {Simone}\ \bibnamefont + {Severini}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Learning hard + quantum distributions with variational autoencoders},}\ }\href {\doibase + 10.1038/s41534-018-0077-z} {\bibfield {journal} {\bibinfo {journal} {npj + Quantum Information}\ }\textbf {\bibinfo {volume} {4}},\ \bibinfo {pages} + {28} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Torlai}\ and\ \citenamefont + {Melko}(2018)}]{Torlai_latent}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Giacomo}\ \bibnamefont + {Torlai}}\ and\ \bibinfo {author} {\bibfnamefont {Roger~G.}\ \bibnamefont + {Melko}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Latent space + purification via neural density operators},}\ }\href {\doibase + 10.1103/PhysRevLett.120.240503} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {120}},\ \bibinfo {pages} + {240503} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Quek}}\ \emph {et~al.}(2018)\citenamefont {{Quek}}, + \citenamefont {{Fort}},\ and\ \citenamefont {{Khoon + Ng}}}]{2018arXiv181206693Q}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Yihui}\ \bibnamefont + {{Quek}}}, \bibinfo {author} {\bibfnamefont {Stanislav}\ \bibnamefont + {{Fort}}}, \ and\ \bibinfo {author} {\bibfnamefont {Hui}\ \bibnamefont + {{Khoon Ng}}},\ }\bibfield {title} {\enquote {\bibinfo {title} {{Adaptive + Quantum State Tomography with Neural Networks}},}\ }\href@noop {} {\bibfield + {journal} {\bibinfo {journal} {arXiv e-prints}\ ,\ \bibinfo {eid} + {arXiv:1812.06693}} (\bibinfo {year} {2018})},\ \Eprint + {http://arxiv.org/abs/1812.06693} {arXiv:1812.06693 [quant-ph]} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Carrasquilla}\ \emph {et~al.}(2019)\citenamefont + {Carrasquilla}, \citenamefont {Torlai}, \citenamefont {Melko},\ and\ + \citenamefont {Aolita}}]{carrasquilla_povm}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Juan}\ \bibnamefont + {Carrasquilla}}, \bibinfo {author} {\bibfnamefont {Giacomo}\ \bibnamefont + {Torlai}}, \bibinfo {author} {\bibfnamefont {Roger~G.}\ \bibnamefont + {Melko}}, \ and\ \bibinfo {author} {\bibfnamefont {Leandro}\ \bibnamefont + {Aolita}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Reconstructing + quantum states with generative models},}\ }\href {\doibase + 10.1038/s42256-019-0028-1} {\bibfield {journal} {\bibinfo {journal} {Nature + Machine Intelligence}\ }\textbf {\bibinfo {volume} {1}},\ \bibinfo {pages} + {155--161} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Palmieri}\ \emph {et~al.}(2020)\citenamefont + {Palmieri}, \citenamefont {Kovlakov}, \citenamefont {Bianchi}, \citenamefont + {Yudin}, \citenamefont {Straupe}, \citenamefont {Biamonte},\ and\ + \citenamefont {Kulik}}]{biamonte_qst}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Adriano~Macarone}\ + \bibnamefont {Palmieri}}, \bibinfo {author} {\bibfnamefont {Egor}\ + \bibnamefont {Kovlakov}}, \bibinfo {author} {\bibfnamefont {Federico}\ + \bibnamefont {Bianchi}}, \bibinfo {author} {\bibfnamefont {Dmitry}\ + \bibnamefont {Yudin}}, \bibinfo {author} {\bibfnamefont {Stanislav}\ + \bibnamefont {Straupe}}, \bibinfo {author} {\bibfnamefont {Jacob~D.}\ + \bibnamefont {Biamonte}}, \ and\ \bibinfo {author} {\bibfnamefont {Sergei}\ + \bibnamefont {Kulik}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Experimental neural network enhanced quantum tomography},}\ }\href {\doibase + 10.1038/s41534-020-0248-6} {\bibfield {journal} {\bibinfo {journal} {npj + Quantum Information}\ }\textbf {\bibinfo {volume} {6}},\ \bibinfo {pages} + {20} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Torlai}\ \emph {et~al.}(2019)\citenamefont {Torlai}, + \citenamefont {Timar}, \citenamefont {van Nieuwenburg}, \citenamefont + {Levine}, \citenamefont {Omran}, \citenamefont {Keesling}, \citenamefont + {Bernien}, \citenamefont {Greiner}, \citenamefont + {Vuleti\ifmmode~\acute{c}\else \'{c}\fi{}}, \citenamefont {Lukin}, + \citenamefont {Melko},\ and\ \citenamefont {Endres}}]{torlai_rydberg19}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Giacomo}\ \bibnamefont + {Torlai}}, \bibinfo {author} {\bibfnamefont {Brian}\ \bibnamefont {Timar}}, + \bibinfo {author} {\bibfnamefont {Evert P.~L.}\ \bibnamefont {van + Nieuwenburg}}, \bibinfo {author} {\bibfnamefont {Harry}\ \bibnamefont + {Levine}}, \bibinfo {author} {\bibfnamefont {Ahmed}\ \bibnamefont {Omran}}, + \bibinfo {author} {\bibfnamefont {Alexander}\ \bibnamefont {Keesling}}, + \bibinfo {author} {\bibfnamefont {Hannes}\ \bibnamefont {Bernien}}, \bibinfo + {author} {\bibfnamefont {Markus}\ \bibnamefont {Greiner}}, \bibinfo {author} + {\bibfnamefont {Vladan}\ \bibnamefont {Vuleti\ifmmode~\acute{c}\else + \'{c}\fi{}}}, \bibinfo {author} {\bibfnamefont {Mikhail~D.}\ \bibnamefont + {Lukin}}, \bibinfo {author} {\bibfnamefont {Roger~G.}\ \bibnamefont {Melko}}, + \ and\ \bibinfo {author} {\bibfnamefont {Manuel}\ \bibnamefont {Endres}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Integrating neural networks + with a quantum simulator for state reconstruction},}\ }\href {\doibase + 10.1103/PhysRevLett.123.230504} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {123}},\ \bibinfo {pages} + {230504} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Xin}\ \emph {et~al.}(2019)\citenamefont {Xin}, + \citenamefont {Lu}, \citenamefont {Cao}, \citenamefont {Anikeeva}, + \citenamefont {Lu}, \citenamefont {Li}, \citenamefont {Long},\ and\ + \citenamefont {Zeng}}]{xin_local-measurement-based_2019}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Tao}\ \bibnamefont + {Xin}}, \bibinfo {author} {\bibfnamefont {Sirui}\ \bibnamefont {Lu}}, + \bibinfo {author} {\bibfnamefont {Ningping}\ \bibnamefont {Cao}}, \bibinfo + {author} {\bibfnamefont {Galit}\ \bibnamefont {Anikeeva}}, \bibinfo {author} + {\bibfnamefont {Dawei}\ \bibnamefont {Lu}}, \bibinfo {author} {\bibfnamefont + {Jun}\ \bibnamefont {Li}}, \bibinfo {author} {\bibfnamefont {Guilu}\ + \bibnamefont {Long}}, \ and\ \bibinfo {author} {\bibfnamefont {Bei}\ + \bibnamefont {Zeng}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Local-measurement-based quantum state tomography via neural networks},}\ + }\href {\doibase 10.1038/s41534-019-0222-3} {\bibfield {journal} {\bibinfo + {journal} {npj Quantum Information}\ }\textbf {\bibinfo {volume} {5}},\ + \bibinfo {pages} {109} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Sehayek}\ \emph {et~al.}(2019)\citenamefont + {Sehayek}, \citenamefont {Golubeva}, \citenamefont {Albergo}, \citenamefont + {Kulchytskyy}, \citenamefont {Torlai},\ and\ \citenamefont + {Melko}}]{Sehayek2019}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Dan}\ \bibnamefont + {Sehayek}}, \bibinfo {author} {\bibfnamefont {Anna}\ \bibnamefont + {Golubeva}}, \bibinfo {author} {\bibfnamefont {Michael~S.}\ \bibnamefont + {Albergo}}, \bibinfo {author} {\bibfnamefont {Bohdan}\ \bibnamefont + {Kulchytskyy}}, \bibinfo {author} {\bibfnamefont {Giacomo}\ \bibnamefont + {Torlai}}, \ and\ \bibinfo {author} {\bibfnamefont {Roger~G.}\ \bibnamefont + {Melko}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Learnability + scaling of quantum states: Restricted boltzmann machines},}\ }\href {\doibase + 10.1103/PhysRevB.100.195125} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {100}},\ \bibinfo {pages} + {195125} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Torlai}\ \emph {et~al.}(2020)\citenamefont {Torlai}, + \citenamefont {Mazzola}, \citenamefont {Carleo},\ and\ \citenamefont + {Mezzacapo}}]{torlai_chemistry}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Giacomo}\ \bibnamefont + {Torlai}}, \bibinfo {author} {\bibfnamefont {Guglielmo}\ \bibnamefont + {Mazzola}}, \bibinfo {author} {\bibfnamefont {Giuseppe}\ \bibnamefont + {Carleo}}, \ and\ \bibinfo {author} {\bibfnamefont {Antonio}\ \bibnamefont + {Mezzacapo}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Precise + measurement of quantum observables with neural-network estimators},}\ }\href + {\doibase 10.1103/PhysRevResearch.2.022060} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. Research}\ }\textbf {\bibinfo {volume} {2}},\ \bibinfo + {pages} {022060} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Melkani}\ \emph {et~al.}(2020)\citenamefont + {Melkani}, \citenamefont {Gneiting},\ and\ \citenamefont + {Nori}}]{PhysRevA.102.022412}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Abhijeet}\ + \bibnamefont {Melkani}}, \bibinfo {author} {\bibfnamefont {Clemens}\ + \bibnamefont {Gneiting}}, \ and\ \bibinfo {author} {\bibfnamefont {Franco}\ + \bibnamefont {Nori}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Eigenstate extraction with neural-network tomography},}\ }\href {\doibase + 10.1103/PhysRevA.102.022412} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {102}},\ \bibinfo {pages} + {022412} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Ahmed}}\ \emph {et~al.}(2020)\citenamefont + {{Ahmed}}, \citenamefont {{S{\'a}nchez Mu{\~n}oz}}, \citenamefont {{Nori}},\ + and\ \citenamefont {{Frisk Kockum}}}]{NoriGAN}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Shahnawaz}\ + \bibnamefont {{Ahmed}}}, \bibinfo {author} {\bibfnamefont {Carlos}\ + \bibnamefont {{S{\'a}nchez Mu{\~n}oz}}}, \bibinfo {author} {\bibfnamefont + {Franco}\ \bibnamefont {{Nori}}}, \ and\ \bibinfo {author} {\bibfnamefont + {Anton}\ \bibnamefont {{Frisk Kockum}}},\ }\bibfield {title} {\enquote + {\bibinfo {title} {{Quantum State Tomography with Conditional Generative + Adversarial Networks}},}\ }\href@noop {} {\bibfield {journal} {\bibinfo + {journal} {arXiv e-prints}\ ,\ \bibinfo {eid} {arXiv:2008.03240}} (\bibinfo + {year} {2020})},\ \Eprint {http://arxiv.org/abs/2008.03240} {arXiv:2008.03240 + [quant-ph]} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Tiunov}\ \emph {et~al.}(2020)\citenamefont {Tiunov}, + \citenamefont {(Vyborova)}, \citenamefont {Ulanov}, \citenamefont {Lvovsky},\ + and\ \citenamefont {Fedorov}}]{Tiunov:20}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.~S.}\ \bibnamefont + {Tiunov}}, \bibinfo {author} {\bibfnamefont {V.~V.~Tiunova}\ \bibnamefont + {(Vyborova)}}, \bibinfo {author} {\bibfnamefont {A.~E.}\ \bibnamefont + {Ulanov}}, \bibinfo {author} {\bibfnamefont {A.~I.}\ \bibnamefont {Lvovsky}}, + \ and\ \bibinfo {author} {\bibfnamefont {A.~K.}\ \bibnamefont {Fedorov}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Experimental quantum + homodyne tomography via machine learning},}\ }\href {\doibase + 10.1364/OPTICA.389482} {\bibfield {journal} {\bibinfo {journal} {Optica}\ + }\textbf {\bibinfo {volume} {7}},\ \bibinfo {pages} {448--454} (\bibinfo + {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Cha}}\ \emph {et~al.}(2020)\citenamefont {{Cha}}, + \citenamefont {{Ginsparg}}, \citenamefont {{Wu}}, \citenamefont + {{Carrasquilla}}, \citenamefont {{McMahon}},\ and\ \citenamefont + {{Kim}}}]{Cha2020}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Peter}\ \bibnamefont + {{Cha}}}, \bibinfo {author} {\bibfnamefont {Paul}\ \bibnamefont + {{Ginsparg}}}, \bibinfo {author} {\bibfnamefont {Felix}\ \bibnamefont + {{Wu}}}, \bibinfo {author} {\bibfnamefont {Juan}\ \bibnamefont + {{Carrasquilla}}}, \bibinfo {author} {\bibfnamefont {Peter~L.}\ \bibnamefont + {{McMahon}}}, \ and\ \bibinfo {author} {\bibfnamefont {Eun-Ah}\ \bibnamefont + {{Kim}}},\ }\bibfield {title} {\enquote {\bibinfo {title} {{Attention-based + Quantum Tomography}},}\ }\href@noop {} {\bibfield {journal} {\bibinfo + {journal} {arXiv e-prints}\ ,\ \bibinfo {eid} {arXiv:2006.12469}} (\bibinfo + {year} {2020})},\ \Eprint {http://arxiv.org/abs/2006.12469} {arXiv:2006.12469 + [quant-ph]} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Neugebauer}\ \emph {et~al.}(2020)\citenamefont + {Neugebauer}, \citenamefont {Fischer}, \citenamefont {J\"ager}, \citenamefont + {Czischek}, \citenamefont {Jochim}, \citenamefont {Weidem\"uller},\ and\ + \citenamefont {G\"arttner}}]{PhysRevA.102.042604}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Marcel}\ \bibnamefont + {Neugebauer}}, \bibinfo {author} {\bibfnamefont {Laurin}\ \bibnamefont + {Fischer}}, \bibinfo {author} {\bibfnamefont {Alexander}\ \bibnamefont + {J\"ager}}, \bibinfo {author} {\bibfnamefont {Stefanie}\ \bibnamefont + {Czischek}}, \bibinfo {author} {\bibfnamefont {Selim}\ \bibnamefont + {Jochim}}, \bibinfo {author} {\bibfnamefont {Matthias}\ \bibnamefont + {Weidem\"uller}}, \ and\ \bibinfo {author} {\bibfnamefont {Martin}\ + \bibnamefont {G\"arttner}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Neural-network quantum state tomography in a two-qubit experiment},}\ }\href + {\doibase 10.1103/PhysRevA.102.042604} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {102}},\ \bibinfo + {pages} {042604} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {De~Vlugt}\ \emph {et~al.}(2020)\citenamefont + {De~Vlugt}, \citenamefont {Iouchtchenko}, \citenamefont {Merali}, + \citenamefont {Roy},\ and\ \citenamefont {Melko}}]{DeVlugt2020}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Isaac J.~S.}\ + \bibnamefont {De~Vlugt}}, \bibinfo {author} {\bibfnamefont {Dmitri}\ + \bibnamefont {Iouchtchenko}}, \bibinfo {author} {\bibfnamefont {Ejaaz}\ + \bibnamefont {Merali}}, \bibinfo {author} {\bibfnamefont {Pierre-Nicholas}\ + \bibnamefont {Roy}}, \ and\ \bibinfo {author} {\bibfnamefont {Roger~G.}\ + \bibnamefont {Melko}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Reconstructing quantum molecular rotor ground states},}\ }\href {\doibase + 10.1103/PhysRevB.102.035108} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {102}},\ \bibinfo {pages} + {035108} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Smith}}\ \emph {et~al.}(2020)\citenamefont + {{Smith}}, \citenamefont {{Gray}},\ and\ \citenamefont + {{Kim}}}]{2020arXiv200907601S}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Alistair W.~R.}\ + \bibnamefont {{Smith}}}, \bibinfo {author} {\bibfnamefont {Johnnie}\ + \bibnamefont {{Gray}}}, \ and\ \bibinfo {author} {\bibfnamefont {M.~S.}\ + \bibnamefont {{Kim}}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {{Efficient Approximate Quantum State Tomography with Basis Dependent + Neural-Networks}},}\ }\href@noop {} {\bibfield {journal} {\bibinfo + {journal} {arXiv e-prints}\ ,\ \bibinfo {eid} {arXiv:2009.07601}} (\bibinfo + {year} {2020})},\ \Eprint {http://arxiv.org/abs/2009.07601} {arXiv:2009.07601 + [quant-ph]} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Torlai}}\ \emph {et~al.}(2020)\citenamefont + {{Torlai}}, \citenamefont {{Wood}}, \citenamefont {{Acharya}}, \citenamefont + {{Carleo}}, \citenamefont {{Carrasquilla}},\ and\ \citenamefont + {{Aolita}}}]{torlai_QPT}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Giacomo}\ \bibnamefont + {{Torlai}}}, \bibinfo {author} {\bibfnamefont {Christopher~J.}\ \bibnamefont + {{Wood}}}, \bibinfo {author} {\bibfnamefont {Atithi}\ \bibnamefont + {{Acharya}}}, \bibinfo {author} {\bibfnamefont {Giuseppe}\ \bibnamefont + {{Carleo}}}, \bibinfo {author} {\bibfnamefont {Juan}\ \bibnamefont + {{Carrasquilla}}}, \ and\ \bibinfo {author} {\bibfnamefont {Leandro}\ + \bibnamefont {{Aolita}}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {{Quantum process tomography with unsupervised learning and tensor + networks}},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} + {arXiv e-prints}\ ,\ \bibinfo {eid} {arXiv:2006.02424}} (\bibinfo {year} + {2020})},\ \Eprint {http://arxiv.org/abs/2006.02424} {arXiv:2006.02424 + [quant-ph]} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Morawetz}}\ \emph {et~al.}(2020)\citenamefont + {{Morawetz}}, \citenamefont {{De Vlugt}}, \citenamefont {{Carrasquilla}},\ + and\ \citenamefont {{Melko}}}]{morawetz2020}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Stewart}\ \bibnamefont + {{Morawetz}}}, \bibinfo {author} {\bibfnamefont {Isaac J.~S.}\ \bibnamefont + {{De Vlugt}}}, \bibinfo {author} {\bibfnamefont {Juan}\ \bibnamefont + {{Carrasquilla}}}, \ and\ \bibinfo {author} {\bibfnamefont {Roger~G.}\ + \bibnamefont {{Melko}}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {{U(1) symmetric recurrent neural networks for quantum state + reconstruction}},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} + {arXiv e-prints}\ ,\ \bibinfo {eid} {arXiv:2010.14514}} (\bibinfo {year} + {2020})},\ \Eprint {http://arxiv.org/abs/2010.14514} {arXiv:2010.14514 + [quant-ph]} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Ahmed}\ \emph {et~al.}(2020)\citenamefont {Ahmed}, + \citenamefont {Munoz}, \citenamefont {Nori},\ and\ \citenamefont + {Kockum}}]{Nori2020}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Shahnawaz}\ + \bibnamefont {Ahmed}}, \bibinfo {author} {\bibfnamefont {Carlos~Sánchez}\ + \bibnamefont {Munoz}}, \bibinfo {author} {\bibfnamefont {Franco}\ + \bibnamefont {Nori}}, \ and\ \bibinfo {author} {\bibfnamefont {Anton~Frisk}\ + \bibnamefont {Kockum}},\ }\href@noop {} {\enquote {\bibinfo {title} + {Classification and reconstruction of optical quantum states with deep neural + networks},}\ } (\bibinfo {year} {2020}),\ \Eprint + {http://arxiv.org/abs/2012.02185} {arXiv:2012.02185 [quant-ph]} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Snyder}\ \emph {et~al.}(2012)\citenamefont {Snyder}, + \citenamefont {Rupp}, \citenamefont {Hansen}, \citenamefont {M\"uller},\ and\ + \citenamefont {Burke}}]{PhysRevLett.108.253002}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {John~C.}\ \bibnamefont + {Snyder}}, \bibinfo {author} {\bibfnamefont {Matthias}\ \bibnamefont {Rupp}}, + \bibinfo {author} {\bibfnamefont {Katja}\ \bibnamefont {Hansen}}, \bibinfo + {author} {\bibfnamefont {Klaus-Robert}\ \bibnamefont {M\"uller}}, \ and\ + \bibinfo {author} {\bibfnamefont {Kieron}\ \bibnamefont {Burke}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Finding density functionals with + machine learning},}\ }\href {\doibase 10.1103/PhysRevLett.108.253002} + {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf + {\bibinfo {volume} {108}},\ \bibinfo {pages} {253002} (\bibinfo {year} + {2012})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Snyder}\ \emph {et~al.}(2013)\citenamefont {Snyder}, + \citenamefont {Rupp}, \citenamefont {Hansen}, \citenamefont {Blooston}, + \citenamefont {Müller},\ and\ \citenamefont + {Burke}}]{doi:10.1063/1.4834075}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {John~C.}\ \bibnamefont + {Snyder}}, \bibinfo {author} {\bibfnamefont {Matthias}\ \bibnamefont {Rupp}}, + \bibinfo {author} {\bibfnamefont {Katja}\ \bibnamefont {Hansen}}, \bibinfo + {author} {\bibfnamefont {Leo}\ \bibnamefont {Blooston}}, \bibinfo {author} + {\bibfnamefont {Klaus-Robert}\ \bibnamefont {Müller}}, \ and\ \bibinfo + {author} {\bibfnamefont {Kieron}\ \bibnamefont {Burke}},\ }\bibfield {title} + {\enquote {\bibinfo {title} {Orbital-free bond breaking via machine + learning},}\ }\href {\doibase 10.1063/1.4834075} {\bibfield {journal} + {\bibinfo {journal} {The Journal of Chemical Physics}\ }\textbf {\bibinfo + {volume} {139}},\ \bibinfo {pages} {224104} (\bibinfo {year} {2013})},\ + \Eprint {http://arxiv.org/abs/https://doi.org/10.1063/1.4834075} + {https://doi.org/10.1063/1.4834075} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Li}\ \emph {et~al.}(2016)\citenamefont {Li}, + \citenamefont {Baker}, \citenamefont {White},\ and\ \citenamefont + {Burke}}]{PhysRevB.94.245129}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Li}~\bibnamefont + {Li}}, \bibinfo {author} {\bibfnamefont {Thomas~E.}\ \bibnamefont {Baker}}, + \bibinfo {author} {\bibfnamefont {Steven~R.}\ \bibnamefont {White}}, \ and\ + \bibinfo {author} {\bibfnamefont {Kieron}\ \bibnamefont {Burke}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Pure density functional for strong + correlation and the thermodynamic limit from machine learning},}\ }\href + {\doibase 10.1103/PhysRevB.94.245129} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {94}},\ \bibinfo + {pages} {245129} (\bibinfo {year} {2016})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Brockherde}\ \emph {et~al.}(2017)\citenamefont + {Brockherde}, \citenamefont {Vogt}, \citenamefont {Li}, \citenamefont + {Tuckerman}, \citenamefont {Burke},\ and\ \citenamefont + {Müller}}]{brockherde_bypassing_2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Felix}\ \bibnamefont + {Brockherde}}, \bibinfo {author} {\bibfnamefont {Leslie}\ \bibnamefont + {Vogt}}, \bibinfo {author} {\bibfnamefont {Li}~\bibnamefont {Li}}, \bibinfo + {author} {\bibfnamefont {Mark~E.}\ \bibnamefont {Tuckerman}}, \bibinfo + {author} {\bibfnamefont {Kieron}\ \bibnamefont {Burke}}, \ and\ \bibinfo + {author} {\bibfnamefont {Klaus-Robert}\ \bibnamefont {Müller}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Bypassing the {Kohn}-{Sham} equations + with machine learning},}\ }\href {\doibase 10.1038/s41467-017-00839-3} + {\bibfield {journal} {\bibinfo {journal} {Nature Communications}\ }\textbf + {\bibinfo {volume} {8}},\ \bibinfo {pages} {872} (\bibinfo {year} + {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Ryczko}\ \emph {et~al.}(2019)\citenamefont {Ryczko}, + \citenamefont {Strubbe},\ and\ \citenamefont + {Tamblyn}}]{PhysRevA.100.022512}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Kevin}\ \bibnamefont + {Ryczko}}, \bibinfo {author} {\bibfnamefont {David~A.}\ \bibnamefont + {Strubbe}}, \ and\ \bibinfo {author} {\bibfnamefont {Isaac}\ \bibnamefont + {Tamblyn}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Deep learning + and density-functional theory},}\ }\href {\doibase + 10.1103/PhysRevA.100.022512} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {100}},\ \bibinfo {pages} + {022512} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Moreno}\ \emph {et~al.}(2020)\citenamefont {Moreno}, + \citenamefont {Carleo},\ and\ \citenamefont + {Georges}}]{PhysRevLett.125.076402}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Javier~Robledo}\ + \bibnamefont {Moreno}}, \bibinfo {author} {\bibfnamefont {Giuseppe}\ + \bibnamefont {Carleo}}, \ and\ \bibinfo {author} {\bibfnamefont {Antoine}\ + \bibnamefont {Georges}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Deep learning the hohenberg-kohn maps of density functional theory},}\ + }\href {\doibase 10.1103/PhysRevLett.125.076402} {\bibfield {journal} + {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {125}},\ + \bibinfo {pages} {076402} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Denner}\ \emph {et~al.}(2020)\citenamefont {Denner}, + \citenamefont {Fischer},\ and\ \citenamefont + {Neupert}}]{PhysRevResearch.2.033388}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~Michael}\ + \bibnamefont {Denner}}, \bibinfo {author} {\bibfnamefont {Mark~H.}\ + \bibnamefont {Fischer}}, \ and\ \bibinfo {author} {\bibfnamefont {Titus}\ + \bibnamefont {Neupert}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Efficient learning of a one-dimensional density functional theory},}\ }\href + {\doibase 10.1103/PhysRevResearch.2.033388} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. Research}\ }\textbf {\bibinfo {volume} {2}},\ \bibinfo + {pages} {033388} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Mehta}}\ and\ \citenamefont + {{Schwab}}(2014)}]{2014arXiv1410.3831M}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Pankaj}\ \bibnamefont + {{Mehta}}}\ and\ \bibinfo {author} {\bibfnamefont {David~J.}\ \bibnamefont + {{Schwab}}},\ }\bibfield {title} {\enquote {\bibinfo {title} {{An exact + mapping between the Variational Renormalization Group and Deep Learning}},}\ + }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {arXiv e-prints}\ + ,\ \bibinfo {eid} {arXiv:1410.3831}} (\bibinfo {year} {2014})},\ \Eprint + {http://arxiv.org/abs/1410.3831} {arXiv:1410.3831 [stat.ML]} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Koch-Janusz}\ and\ \citenamefont + {Ringel}(2018)}]{koch-janusz_mutual_2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Maciej}\ \bibnamefont + {Koch-Janusz}}\ and\ \bibinfo {author} {\bibfnamefont {Zohar}\ \bibnamefont + {Ringel}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Mutual + information, neural networks and the renormalization group},}\ }\href + {\doibase 10.1038/s41567-018-0081-4} {\bibfield {journal} {\bibinfo + {journal} {Nature Physics}\ }\textbf {\bibinfo {volume} {14}},\ \bibinfo + {pages} {578--582} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Iso}\ \emph {et~al.}(2018)\citenamefont {Iso}, + \citenamefont {Shiba},\ and\ \citenamefont {Yokoo}}]{PhysRevE.97.053304}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Satoshi}\ \bibnamefont + {Iso}}, \bibinfo {author} {\bibfnamefont {Shotaro}\ \bibnamefont {Shiba}}, \ + and\ \bibinfo {author} {\bibfnamefont {Sumito}\ \bibnamefont {Yokoo}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Scale-invariant feature + extraction of neural network and renormalization group flow},}\ }\href + {\doibase 10.1103/PhysRevE.97.053304} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. E}\ }\textbf {\bibinfo {volume} {97}},\ \bibinfo + {pages} {053304} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Li}\ and\ \citenamefont + {Wang}(2018)}]{PhysRevLett.121.260601}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Shuo-Hui}\ + \bibnamefont {Li}}\ and\ \bibinfo {author} {\bibfnamefont {Lei}\ \bibnamefont + {Wang}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Neural network + renormalization group},}\ }\href {\doibase 10.1103/PhysRevLett.121.260601} + {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf + {\bibinfo {volume} {121}},\ \bibinfo {pages} {260601} (\bibinfo {year} + {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Hu}\ \emph {et~al.}(2020)\citenamefont {Hu}, + \citenamefont {Li}, \citenamefont {Wang},\ and\ \citenamefont + {You}}]{PhysRevResearch.2.023369}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Hong-Ye}\ \bibnamefont + {Hu}}, \bibinfo {author} {\bibfnamefont {Shuo-Hui}\ \bibnamefont {Li}}, + \bibinfo {author} {\bibfnamefont {Lei}\ \bibnamefont {Wang}}, \ and\ \bibinfo + {author} {\bibfnamefont {Yi-Zhuang}\ \bibnamefont {You}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Machine learning holographic mapping by + neural network renormalization group},}\ }\href {\doibase + 10.1103/PhysRevResearch.2.023369} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. Research}\ }\textbf {\bibinfo {volume} {2}},\ \bibinfo {pages} + {023369} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Chung}}\ and\ \citenamefont + {{Kao}}(2020)}]{2020arXiv201005703C}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Jui-Hui}\ \bibnamefont + {{Chung}}}\ and\ \bibinfo {author} {\bibfnamefont {Ying-Jer}\ \bibnamefont + {{Kao}}},\ }\bibfield {title} {\enquote {\bibinfo {title} {{Neural Monte + Carlo Renormalization Group}},}\ }\href@noop {} {\bibfield {journal} + {\bibinfo {journal} {arXiv e-prints}\ ,\ \bibinfo {eid} {arXiv:2010.05703}} + (\bibinfo {year} {2020})},\ \Eprint {http://arxiv.org/abs/2010.05703} + {arXiv:2010.05703 [cond-mat.dis-nn]} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Torlai}\ and\ \citenamefont + {Melko}(2017)}]{torlai_neural_2016}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Giacomo}\ \bibnamefont + {Torlai}}\ and\ \bibinfo {author} {\bibfnamefont {Roger~G.}\ \bibnamefont + {Melko}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Neural decoder + for topological codes},}\ }\href {\doibase 10.1103/PhysRevLett.119.030501} + {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf + {\bibinfo {volume} {119}},\ \bibinfo {pages} {030501} (\bibinfo {year} + {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Krastanov}\ and\ \citenamefont + {Jiang}(2017)}]{krastanov_deep_2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Stefan}\ \bibnamefont + {Krastanov}}\ and\ \bibinfo {author} {\bibfnamefont {Liang}\ \bibnamefont + {Jiang}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Deep {Neural} + {Network} {Probabilistic} {Decoder} for {Stabilizer} {Codes}},}\ }\href + {\doibase 10.1038/s41598-017-11266-1} {\bibfield {journal} {\bibinfo + {journal} {Scientific Reports}\ }\textbf {\bibinfo {volume} {7}},\ \bibinfo + {pages} {11003} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Varsamopoulos}\ \emph {et~al.}(2017)\citenamefont + {Varsamopoulos}, \citenamefont {Criger},\ and\ \citenamefont + {Bertels}}]{Varsamopoulos_2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Savvas}\ \bibnamefont + {Varsamopoulos}}, \bibinfo {author} {\bibfnamefont {Ben}\ \bibnamefont + {Criger}}, \ and\ \bibinfo {author} {\bibfnamefont {Koen}\ \bibnamefont + {Bertels}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Decoding small + surface codes with feedforward neural networks},}\ }\href {\doibase + 10.1088/2058-9565/aa955a} {\bibfield {journal} {\bibinfo {journal} {Quantum + Science and Technology}\ }\textbf {\bibinfo {volume} {3}},\ \bibinfo {pages} + {015004} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Baireuther}\ \emph {et~al.}(2018)\citenamefont + {Baireuther}, \citenamefont {O'Brien}, \citenamefont {Tarasinski},\ and\ + \citenamefont {Beenakker}}]{Baireuther2018machinelearning}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Paul}\ \bibnamefont + {Baireuther}}, \bibinfo {author} {\bibfnamefont {Thomas~E.}\ \bibnamefont + {O'Brien}}, \bibinfo {author} {\bibfnamefont {Brian}\ \bibnamefont + {Tarasinski}}, \ and\ \bibinfo {author} {\bibfnamefont {Carlo W.~J.}\ + \bibnamefont {Beenakker}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Machine-learning-assisted correction of correlated qubit errors in a + topological code},}\ }\href {\doibase 10.22331/q-2018-01-29-48} {\bibfield + {journal} {\bibinfo {journal} {{Quantum}}\ }\textbf {\bibinfo {volume} + {2}},\ \bibinfo {pages} {48} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Chamberland}\ and\ \citenamefont + {Ronagh}(2018)}]{Chamberland_2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Christopher}\ + \bibnamefont {Chamberland}}\ and\ \bibinfo {author} {\bibfnamefont {Pooya}\ + \bibnamefont {Ronagh}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Deep neural decoders for near term fault-tolerant experiments},}\ }\href + {\doibase 10.1088/2058-9565/aad1f7} {\bibfield {journal} {\bibinfo + {journal} {Quantum Science and Technology}\ }\textbf {\bibinfo {volume} + {3}},\ \bibinfo {pages} {044002} (\bibinfo {year} {2018})}\BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Breuckmann}\ and\ \citenamefont + {Ni}(2018)}]{Breuckmann2018scalableneural}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Nikolas~P.}\ + \bibnamefont {Breuckmann}}\ and\ \bibinfo {author} {\bibfnamefont {Xiaotong}\ + \bibnamefont {Ni}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Scalable {N}eural {N}etwork {D}ecoders for {H}igher {D}imensional {Q}uantum + {C}odes},}\ }\href {\doibase 10.22331/q-2018-05-24-68} {\bibfield {journal} + {\bibinfo {journal} {{Quantum}}\ }\textbf {\bibinfo {volume} {2}},\ \bibinfo + {pages} {68} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Nautrup}\ \emph {et~al.}(2019)\citenamefont + {Nautrup}, \citenamefont {Delfosse}, \citenamefont {Dunjko}, \citenamefont + {Briegel},\ and\ \citenamefont {Friis}}]{Nautrup2019optimizingquantum}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Hendrik~Poulsen}\ + \bibnamefont {Nautrup}}, \bibinfo {author} {\bibfnamefont {Nicolas}\ + \bibnamefont {Delfosse}}, \bibinfo {author} {\bibfnamefont {Vedran}\ + \bibnamefont {Dunjko}}, \bibinfo {author} {\bibfnamefont {Hans~J.}\ + \bibnamefont {Briegel}}, \ and\ \bibinfo {author} {\bibfnamefont {Nicolai}\ + \bibnamefont {Friis}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Optimizing {Q}uantum {E}rror {C}orrection {C}odes with {R}einforcement + {L}earning},}\ }\href {\doibase 10.22331/q-2019-12-16-215} {\bibfield + {journal} {\bibinfo {journal} {{Quantum}}\ }\textbf {\bibinfo {volume} + {3}},\ \bibinfo {pages} {215} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Liu}\ and\ \citenamefont + {Poulin}(2019)}]{PhysRevLett.122.200501}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Ye-Hua}\ \bibnamefont + {Liu}}\ and\ \bibinfo {author} {\bibfnamefont {David}\ \bibnamefont + {Poulin}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Neural + belief-propagation decoders for quantum error-correcting codes},}\ }\href + {\doibase 10.1103/PhysRevLett.122.200501} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {122}},\ \bibinfo + {pages} {200501} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Maskara}\ \emph {et~al.}(2019)\citenamefont + {Maskara}, \citenamefont {Kubica},\ and\ \citenamefont + {Jochym-O'Connor}}]{PhysRevA.99.052351}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Nishad}\ \bibnamefont + {Maskara}}, \bibinfo {author} {\bibfnamefont {Aleksander}\ \bibnamefont + {Kubica}}, \ and\ \bibinfo {author} {\bibfnamefont {Tomas}\ \bibnamefont + {Jochym-O'Connor}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Advantages of versatile neural-network decoding for topological codes},}\ + }\href {\doibase 10.1103/PhysRevA.99.052351} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {99}},\ \bibinfo + {pages} {052351} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Andreasson}\ \emph {et~al.}(2019)\citenamefont + {Andreasson}, \citenamefont {Johansson}, \citenamefont {Liljestrand},\ and\ + \citenamefont {Granath}}]{Andreasson2019quantumerror}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Philip}\ \bibnamefont + {Andreasson}}, \bibinfo {author} {\bibfnamefont {Joel}\ \bibnamefont + {Johansson}}, \bibinfo {author} {\bibfnamefont {Simon}\ \bibnamefont + {Liljestrand}}, \ and\ \bibinfo {author} {\bibfnamefont {Mats}\ \bibnamefont + {Granath}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum error + correction for the toric code using deep reinforcement learning},}\ }\href + {\doibase 10.22331/q-2019-09-02-183} {\bibfield {journal} {\bibinfo + {journal} {{Quantum}}\ }\textbf {\bibinfo {volume} {3}},\ \bibinfo {pages} + {183} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Sweke}\ \emph {et~al.}(2020)\citenamefont {Sweke}, + \citenamefont {Kesselring}, \citenamefont {van Nieuwenburg},\ and\ + \citenamefont {Eisert}}]{Evert2020QEC}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Ryan}\ \bibnamefont + {Sweke}}, \bibinfo {author} {\bibfnamefont {Markus}\ \bibnamefont + {Kesselring}}, \bibinfo {author} {\bibfnamefont {Evert}\ \bibnamefont {van + Nieuwenburg}}, \ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont + {Eisert}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Reinforcement + learning decoders for fault-tolerant quantum computation},}\ }\href + {http://iopscience.iop.org/article/10.1088/2632-2153/abc609} {\bibfield + {journal} {\bibinfo {journal} {Machine Learning: Science and Technology}\ } + (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Ni}(2020)}]{Ni2020neuralnetwork}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Xiaotong}\ + \bibnamefont {Ni}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Neural + {N}etwork {D}ecoders for {L}arge-{D}istance 2{D} {T}oric {C}odes},}\ }\href + {\doibase 10.22331/q-2020-08-24-310} {\bibfield {journal} {\bibinfo + {journal} {{Quantum}}\ }\textbf {\bibinfo {volume} {4}},\ \bibinfo {pages} + {310} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Davaasuren}\ \emph {et~al.}(2020)\citenamefont + {Davaasuren}, \citenamefont {Suzuki}, \citenamefont {Fujii},\ and\ + \citenamefont {Koashi}}]{PhysRevResearch.2.033399}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Amarsanaa}\ + \bibnamefont {Davaasuren}}, \bibinfo {author} {\bibfnamefont {Yasunari}\ + \bibnamefont {Suzuki}}, \bibinfo {author} {\bibfnamefont {Keisuke}\ + \bibnamefont {Fujii}}, \ and\ \bibinfo {author} {\bibfnamefont {Masato}\ + \bibnamefont {Koashi}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {General framework for constructing fast and near-optimal + machine-learning-based decoder of the topological stabilizer codes},}\ }\href + {\doibase 10.1103/PhysRevResearch.2.033399} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. Research}\ }\textbf {\bibinfo {volume} {2}},\ \bibinfo + {pages} {033399} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Fitzek}\ \emph {et~al.}(2020)\citenamefont {Fitzek}, + \citenamefont {Eliasson}, \citenamefont {Kockum},\ and\ \citenamefont + {Granath}}]{PhysRevResearch.2.023230}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {David}\ \bibnamefont + {Fitzek}}, \bibinfo {author} {\bibfnamefont {Mattias}\ \bibnamefont + {Eliasson}}, \bibinfo {author} {\bibfnamefont {Anton~Frisk}\ \bibnamefont + {Kockum}}, \ and\ \bibinfo {author} {\bibfnamefont {Mats}\ \bibnamefont + {Granath}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Deep q-learning + decoder for depolarizing noise on the toric code},}\ }\href {\doibase + 10.1103/PhysRevResearch.2.023230} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. Research}\ }\textbf {\bibinfo {volume} {2}},\ \bibinfo {pages} + {023230} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {You}\ \emph {et~al.}(2018)\citenamefont {You}, + \citenamefont {Yang},\ and\ \citenamefont {Qi}}]{PhysRevB.97.045153}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Yi-Zhuang}\ + \bibnamefont {You}}, \bibinfo {author} {\bibfnamefont {Zhao}\ \bibnamefont + {Yang}}, \ and\ \bibinfo {author} {\bibfnamefont {Xiao-Liang}\ \bibnamefont + {Qi}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Machine learning + spatial geometry from entanglement features},}\ }\href {\doibase + 10.1103/PhysRevB.97.045153} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. B}\ }\textbf {\bibinfo {volume} {97}},\ \bibinfo {pages} {045153} + (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Seif}\ \emph {et~al.}(2018)\citenamefont {Seif}, + \citenamefont {Landsman}, \citenamefont {Linke}, \citenamefont {Figgatt}, + \citenamefont {Monroe},\ and\ \citenamefont {Hafezi}}]{Seif_2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Alireza}\ \bibnamefont + {Seif}}, \bibinfo {author} {\bibfnamefont {Kevin~A}\ \bibnamefont + {Landsman}}, \bibinfo {author} {\bibfnamefont {Norbert~M}\ \bibnamefont + {Linke}}, \bibinfo {author} {\bibfnamefont {Caroline}\ \bibnamefont + {Figgatt}}, \bibinfo {author} {\bibfnamefont {C}~\bibnamefont {Monroe}}, \ + and\ \bibinfo {author} {\bibfnamefont {Mohammad}\ \bibnamefont {Hafezi}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Machine learning assisted + readout of trapped-ion qubits},}\ }\href {\doibase 10.1088/1361-6455/aad62b} + {\bibfield {journal} {\bibinfo {journal} {Journal of Physics B: Atomic, + Molecular and Optical Physics}\ }\textbf {\bibinfo {volume} {51}},\ \bibinfo + {pages} {174006} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Melnikov}\ \emph {et~al.}(2018)\citenamefont + {Melnikov}, \citenamefont {Poulsen~Nautrup}, \citenamefont {Krenn}, + \citenamefont {Dunjko}, \citenamefont {Tiersch}, \citenamefont {Zeilinger},\ + and\ \citenamefont {Briegel}}]{Melnikov1221}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Alexey~A.}\ + \bibnamefont {Melnikov}}, \bibinfo {author} {\bibfnamefont {Hendrik}\ + \bibnamefont {Poulsen~Nautrup}}, \bibinfo {author} {\bibfnamefont {Mario}\ + \bibnamefont {Krenn}}, \bibinfo {author} {\bibfnamefont {Vedran}\ + \bibnamefont {Dunjko}}, \bibinfo {author} {\bibfnamefont {Markus}\ + \bibnamefont {Tiersch}}, \bibinfo {author} {\bibfnamefont {Anton}\ + \bibnamefont {Zeilinger}}, \ and\ \bibinfo {author} {\bibfnamefont {Hans~J.}\ + \bibnamefont {Briegel}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Active learning machine learns to create new quantum experiments},}\ }\href + {\doibase 10.1073/pnas.1714936115} {\bibfield {journal} {\bibinfo {journal} + {Proceedings of the National Academy of Sciences}\ }\textbf {\bibinfo + {volume} {115}},\ \bibinfo {pages} {1221--1226} (\bibinfo {year} {2018})},\ + \Eprint + {http://arxiv.org/abs/https://www.pnas.org/content/115/6/1221.full.pdf} + {https://www.pnas.org/content/115/6/1221.full.pdf} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Dunjko}\ and\ \citenamefont + {Briegel}(2018)}]{Dunjko_2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Vedran}\ \bibnamefont + {Dunjko}}\ and\ \bibinfo {author} {\bibfnamefont {Hans~J}\ \bibnamefont + {Briegel}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Machine + learning {\&} artificial intelligence in the quantum domain: a review of + recent progress},}\ }\href {\doibase 10.1088/1361-6633/aab406} {\bibfield + {journal} {\bibinfo {journal} {Reports on Progress in Physics}\ }\textbf + {\bibinfo {volume} {81}},\ \bibinfo {pages} {074001} (\bibinfo {year} + {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Zhao}\ \emph {et~al.}(2019)\citenamefont {Zhao}, + \citenamefont {Kao}, \citenamefont {Wu},\ and\ \citenamefont + {Kao}}]{PhysRevE.99.062106}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Kai-Wen}\ \bibnamefont + {Zhao}}, \bibinfo {author} {\bibfnamefont {Wen-Han}\ \bibnamefont {Kao}}, + \bibinfo {author} {\bibfnamefont {Kai-Hsin}\ \bibnamefont {Wu}}, \ and\ + \bibinfo {author} {\bibfnamefont {Ying-Jer}\ \bibnamefont {Kao}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Generation of ice states through deep + reinforcement learning},}\ }\href {\doibase 10.1103/PhysRevE.99.062106} + {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. E}\ }\textbf {\bibinfo + {volume} {99}},\ \bibinfo {pages} {062106} (\bibinfo {year} + {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Carrasquilla}}\ \emph {et~al.}(2019)\citenamefont + {{Carrasquilla}}, \citenamefont {{Luo}}, \citenamefont {{P{\'e}rez}}, + \citenamefont {{Milsted}}, \citenamefont {{Clark}}, \citenamefont + {{Volkovs}},\ and\ \citenamefont {{Aolita}}}]{2019arXiv191211052C}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Juan}\ \bibnamefont + {{Carrasquilla}}}, \bibinfo {author} {\bibfnamefont {Di}~\bibnamefont + {{Luo}}}, \bibinfo {author} {\bibfnamefont {Felipe}\ \bibnamefont + {{P{\'e}rez}}}, \bibinfo {author} {\bibfnamefont {Ashley}\ \bibnamefont + {{Milsted}}}, \bibinfo {author} {\bibfnamefont {Bryan~K.}\ \bibnamefont + {{Clark}}}, \bibinfo {author} {\bibfnamefont {Maksims}\ \bibnamefont + {{Volkovs}}}, \ and\ \bibinfo {author} {\bibfnamefont {Leandro}\ \bibnamefont + {{Aolita}}},\ }\bibfield {title} {\enquote {\bibinfo {title} {{Probabilistic + Simulation of Quantum Circuits with the Transformer}},}\ }\href@noop {} + {\bibfield {journal} {\bibinfo {journal} {arXiv e-prints}\ ,\ \bibinfo + {eid} {arXiv:1912.11052}} (\bibinfo {year} {2019})},\ \Eprint + {http://arxiv.org/abs/1912.11052} {arXiv:1912.11052 [cond-mat.str-el]} + \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Efthymiou}\ \emph {et~al.}(2019)\citenamefont + {Efthymiou}, \citenamefont {Beach},\ and\ \citenamefont + {Melko}}]{PhysRevB.99.075113}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Stavros}\ \bibnamefont + {Efthymiou}}, \bibinfo {author} {\bibfnamefont {Matthew J.~S.}\ \bibnamefont + {Beach}}, \ and\ \bibinfo {author} {\bibfnamefont {Roger~G.}\ \bibnamefont + {Melko}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Super-resolving + the ising model with convolutional neural networks},}\ }\href {\doibase + 10.1103/PhysRevB.99.075113} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. B}\ }\textbf {\bibinfo {volume} {99}},\ \bibinfo {pages} {075113} + (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Iten}\ \emph {et~al.}(2020)\citenamefont {Iten}, + \citenamefont {Metger}, \citenamefont {Wilming}, \citenamefont {del Rio},\ + and\ \citenamefont {Renner}}]{PhysRevLett.124.010508}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Raban}\ \bibnamefont + {Iten}}, \bibinfo {author} {\bibfnamefont {Tony}\ \bibnamefont {Metger}}, + \bibinfo {author} {\bibfnamefont {Henrik}\ \bibnamefont {Wilming}}, \bibinfo + {author} {\bibfnamefont {L\'{\i}dia}\ \bibnamefont {del Rio}}, \ and\ + \bibinfo {author} {\bibfnamefont {Renato}\ \bibnamefont {Renner}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Discovering physical + concepts with neural networks},}\ }\href {\doibase + 10.1103/PhysRevLett.124.010508} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {124}},\ \bibinfo {pages} + {010508} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Flurin}\ \emph {et~al.}(2020)\citenamefont {Flurin}, + \citenamefont {Martin}, \citenamefont {Hacohen-Gourgy},\ and\ \citenamefont + {Siddiqi}}]{PhysRevX.10.011006}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont + {Flurin}}, \bibinfo {author} {\bibfnamefont {L.~S.}\ \bibnamefont {Martin}}, + \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Hacohen-Gourgy}}, \ and\ + \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont {Siddiqi}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Using a recurrent neural network to + reconstruct quantum dynamics of a superconducting qubit from physical + observations},}\ }\href {\doibase 10.1103/PhysRevX.10.011006} {\bibfield + {journal} {\bibinfo {journal} {Phys. Rev. X}\ }\textbf {\bibinfo {volume} + {10}},\ \bibinfo {pages} {011006} (\bibinfo {year} {2020})}\BibitemShut + {NoStop}% +\bibitem [{\citenamefont {{Hashimoto}}\ \emph {et~al.}(2020)\citenamefont + {{Hashimoto}}, \citenamefont {{Hu}},\ and\ \citenamefont + {{You}}}]{2020arXiv200600712H}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Koji}\ \bibnamefont + {{Hashimoto}}}, \bibinfo {author} {\bibfnamefont {Hong-Ye}\ \bibnamefont + {{Hu}}}, \ and\ \bibinfo {author} {\bibfnamefont {Yi-Zhuang}\ \bibnamefont + {{You}}},\ }\bibfield {title} {\enquote {\bibinfo {title} {{Neural ODE and + Holographic QCD}},}\ }\href@noop {} {\bibfield {journal} {\bibinfo + {journal} {arXiv e-prints}\ ,\ \bibinfo {eid} {arXiv:2006.00712}} (\bibinfo + {year} {2020})},\ \Eprint {http://arxiv.org/abs/2006.00712} {arXiv:2006.00712 + [hep-th]} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Luo}}\ \emph {et~al.}(2020)\citenamefont {{Luo}}, + \citenamefont {{Chen}}, \citenamefont {{Carrasquilla}},\ and\ \citenamefont + {{Clark}}}]{2020arXiv200905580L}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Di}~\bibnamefont + {{Luo}}}, \bibinfo {author} {\bibfnamefont {Zhuo}\ \bibnamefont {{Chen}}}, + \bibinfo {author} {\bibfnamefont {Juan}\ \bibnamefont {{Carrasquilla}}}, \ + and\ \bibinfo {author} {\bibfnamefont {Bryan~K.}\ \bibnamefont {{Clark}}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {{Autoregressive Neural + Network for Simulating Open Quantum Systems via a Probabilistic + Formulation}},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} + {arXiv e-prints}\ ,\ \bibinfo {eid} {arXiv:2009.05580}} (\bibinfo {year} + {2020})},\ \Eprint {http://arxiv.org/abs/2009.05580} {arXiv:2009.05580 + [cond-mat.str-el]} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Lode}}\ \emph {et~al.}(2020)\citenamefont {{Lode}}, + \citenamefont {{Lin}}, \citenamefont {{B{\"u}ttner}}, \citenamefont + {{Papariello}}, \citenamefont {{L{\'e}v{\^e}que}}, \citenamefont {{Chitra}}, + \citenamefont {{Tsatsos}}, \citenamefont {{Jaksch}},\ and\ \citenamefont + {{Molignini}}}]{2020arXiv201014510L}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Axel U.~J.}\ + \bibnamefont {{Lode}}}, \bibinfo {author} {\bibfnamefont {Rui}\ \bibnamefont + {{Lin}}}, \bibinfo {author} {\bibfnamefont {Miriam}\ \bibnamefont + {{B{\"u}ttner}}}, \bibinfo {author} {\bibfnamefont {Luca}\ \bibnamefont + {{Papariello}}}, \bibinfo {author} {\bibfnamefont {Camille}\ \bibnamefont + {{L{\'e}v{\^e}que}}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont + {{Chitra}}}, \bibinfo {author} {\bibfnamefont {Marios~C.}\ \bibnamefont + {{Tsatsos}}}, \bibinfo {author} {\bibfnamefont {Dieter}\ \bibnamefont + {{Jaksch}}}, \ and\ \bibinfo {author} {\bibfnamefont {Paolo}\ \bibnamefont + {{Molignini}}},\ }\bibfield {title} {\enquote {\bibinfo {title} {{Optimized + Observable Readout from Single-shot Images of Ultracold Atoms via Machine + Learning}},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} + {arXiv e-prints}\ ,\ \bibinfo {eid} {arXiv:2010.14510}} (\bibinfo {year} + {2020})},\ \Eprint {http://arxiv.org/abs/2010.14510} {arXiv:2010.14510 + [cond-mat.quant-gas]} \BibitemShut {NoStop}% +\bibitem [{cod()}]{coderepo}% + \BibitemOpen + \href {https://github.com/GTorlai/NeuralNetworks-for-Quantum} {\enquote + {\bibinfo {title} {https://github.com/gtorlai/neuralnetworks-for-quantum},}\ + }\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Goodfellow}\ \emph {et~al.}(2016)\citenamefont + {Goodfellow}, \citenamefont {Bengio},\ and\ \citenamefont + {Courville}}]{Goodfellow-et-al-2016}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Ian}\ \bibnamefont + {Goodfellow}}, \bibinfo {author} {\bibfnamefont {Yoshua}\ \bibnamefont + {Bengio}}, \ and\ \bibinfo {author} {\bibfnamefont {Aaron}\ \bibnamefont + {Courville}},\ }\href@noop {} {\emph {\bibinfo {title} {Deep Learning}}}\ + (\bibinfo {publisher} {MIT Press},\ \bibinfo {year} {2016})\ \bibinfo {note} + {\url{http://www.deeplearningbook.org}}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Bishop}(2006)}]{10.5555/1162264}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Christopher~M.}\ + \bibnamefont {Bishop}},\ }\href@noop {} {\emph {\bibinfo {title} {Pattern + Recognition and Machine Learning (Information Science and Statistics)}}}\ + (\bibinfo {publisher} {Springer-Verlag},\ \bibinfo {address} {Berlin, + Heidelberg},\ \bibinfo {year} {2006})\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Sutton}\ and\ \citenamefont + {Barto}(1998)}]{10.5555/551283}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Richard~S.}\ + \bibnamefont {Sutton}}\ and\ \bibinfo {author} {\bibfnamefont {Andrew~G.}\ + \bibnamefont {Barto}},\ }\href@noop {} {\emph {\bibinfo {title} {Introduction + to Reinforcement Learning}}},\ \bibinfo {edition} {1st}\ ed.\ (\bibinfo + {publisher} {MIT Press},\ \bibinfo {address} {Cambridge, MA, USA},\ \bibinfo + {year} {1998})\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Mnih}\ \emph {et~al.}(2015)\citenamefont {Mnih}, + \citenamefont {Kavukcuoglu}, \citenamefont {Silver}, \citenamefont {Rusu}, + \citenamefont {Veness}, \citenamefont {Bellemare}, \citenamefont {Graves}, + \citenamefont {Riedmiller}, \citenamefont {Fidjeland}, \citenamefont + {Ostrovski}, \citenamefont {Petersen}, \citenamefont {Beattie}, \citenamefont + {Sadik}, \citenamefont {Antonoglou}, \citenamefont {King}, \citenamefont + {Kumaran}, \citenamefont {Wierstra}, \citenamefont {Legg},\ and\ + \citenamefont {Hassabis}}]{mnih_human-level_2015}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Volodymyr}\ + \bibnamefont {Mnih}}, \bibinfo {author} {\bibfnamefont {Koray}\ \bibnamefont + {Kavukcuoglu}}, \bibinfo {author} {\bibfnamefont {David}\ \bibnamefont + {Silver}}, \bibinfo {author} {\bibfnamefont {Andrei~A.}\ \bibnamefont + {Rusu}}, \bibinfo {author} {\bibfnamefont {Joel}\ \bibnamefont {Veness}}, + \bibinfo {author} {\bibfnamefont {Marc~G.}\ \bibnamefont {Bellemare}}, + \bibinfo {author} {\bibfnamefont {Alex}\ \bibnamefont {Graves}}, \bibinfo + {author} {\bibfnamefont {Martin}\ \bibnamefont {Riedmiller}}, \bibinfo + {author} {\bibfnamefont {Andreas~K.}\ \bibnamefont {Fidjeland}}, \bibinfo + {author} {\bibfnamefont {Georg}\ \bibnamefont {Ostrovski}}, \bibinfo {author} + {\bibfnamefont {Stig}\ \bibnamefont {Petersen}}, \bibinfo {author} + {\bibfnamefont {Charles}\ \bibnamefont {Beattie}}, \bibinfo {author} + {\bibfnamefont {Amir}\ \bibnamefont {Sadik}}, \bibinfo {author} + {\bibfnamefont {Ioannis}\ \bibnamefont {Antonoglou}}, \bibinfo {author} + {\bibfnamefont {Helen}\ \bibnamefont {King}}, \bibinfo {author} + {\bibfnamefont {Dharshan}\ \bibnamefont {Kumaran}}, \bibinfo {author} + {\bibfnamefont {Daan}\ \bibnamefont {Wierstra}}, \bibinfo {author} + {\bibfnamefont {Shane}\ \bibnamefont {Legg}}, \ and\ \bibinfo {author} + {\bibfnamefont {Demis}\ \bibnamefont {Hassabis}},\ }\bibfield {title} + {\enquote {\bibinfo {title} {Human-level control through deep reinforcement + learning},}\ }\href {\doibase 10.1038/nature14236} {\bibfield {journal} + {\bibinfo {journal} {Nature}\ }\textbf {\bibinfo {volume} {518}},\ \bibinfo + {pages} {529--533} (\bibinfo {year} {2015})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Silver}\ \emph {et~al.}(2016)\citenamefont {Silver}, + \citenamefont {Huang}, \citenamefont {Maddison}, \citenamefont {Guez}, + \citenamefont {Sifre}, \citenamefont {{van den Driessche}}, \citenamefont + {Schrittwieser}, \citenamefont {Antonoglou}, \citenamefont {Panneershelvam}, + \citenamefont {Lanctot}, \citenamefont {Dieleman}, \citenamefont {Grewe}, + \citenamefont {Nham}, \citenamefont {Kalchbrenner}, \citenamefont + {Sutskever}, \citenamefont {Lillicrap}, \citenamefont {Leach}, \citenamefont + {Kavukcuoglu}, \citenamefont {Graepel},\ and\ \citenamefont + {Hassabis}}]{silver2016}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {David}\ \bibnamefont + {Silver}}, \bibinfo {author} {\bibfnamefont {Aja}\ \bibnamefont {Huang}}, + \bibinfo {author} {\bibfnamefont {Chris~J.}\ \bibnamefont {Maddison}}, + \bibinfo {author} {\bibfnamefont {Arthur}\ \bibnamefont {Guez}}, \bibinfo + {author} {\bibfnamefont {Laurent}\ \bibnamefont {Sifre}}, \bibinfo {author} + {\bibfnamefont {George}\ \bibnamefont {{van den Driessche}}}, \bibinfo + {author} {\bibfnamefont {Julian}\ \bibnamefont {Schrittwieser}}, \bibinfo + {author} {\bibfnamefont {Ioannis}\ \bibnamefont {Antonoglou}}, \bibinfo + {author} {\bibfnamefont {Veda}\ \bibnamefont {Panneershelvam}}, \bibinfo + {author} {\bibfnamefont {Marc}\ \bibnamefont {Lanctot}}, \bibinfo {author} + {\bibfnamefont {Sander}\ \bibnamefont {Dieleman}}, \bibinfo {author} + {\bibfnamefont {Dominik}\ \bibnamefont {Grewe}}, \bibinfo {author} + {\bibfnamefont {John}\ \bibnamefont {Nham}}, \bibinfo {author} {\bibfnamefont + {Nal}\ \bibnamefont {Kalchbrenner}}, \bibinfo {author} {\bibfnamefont {Ilya}\ + \bibnamefont {Sutskever}}, \bibinfo {author} {\bibfnamefont {Timothy}\ + \bibnamefont {Lillicrap}}, \bibinfo {author} {\bibfnamefont {Madeleine}\ + \bibnamefont {Leach}}, \bibinfo {author} {\bibfnamefont {Koray}\ \bibnamefont + {Kavukcuoglu}}, \bibinfo {author} {\bibfnamefont {Thore}\ \bibnamefont + {Graepel}}, \ and\ \bibinfo {author} {\bibfnamefont {Demis}\ \bibnamefont + {Hassabis}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Mastering the + game of {{Go}} with deep neural networks and tree search},}\ }\href {\doibase + 10.1038/nature16961} {\bibfield {journal} {\bibinfo {journal} {Nature}\ + }\textbf {\bibinfo {volume} {529}},\ \bibinfo {pages} {484--489} (\bibinfo + {year} {2016})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Bukov}\ \emph + {et~al.}(2018{\natexlab{b}})\citenamefont {Bukov}, \citenamefont {Day}, + \citenamefont {Sels}, \citenamefont {Weinberg}, \citenamefont {Polkovnikov},\ + and\ \citenamefont {Mehta}}]{bukov2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Marin}\ \bibnamefont + {Bukov}}, \bibinfo {author} {\bibfnamefont {Alexandre G.~R.}\ \bibnamefont + {Day}}, \bibinfo {author} {\bibfnamefont {Dries}\ \bibnamefont {Sels}}, + \bibinfo {author} {\bibfnamefont {Phillip}\ \bibnamefont {Weinberg}}, + \bibinfo {author} {\bibfnamefont {Anatoli}\ \bibnamefont {Polkovnikov}}, \ + and\ \bibinfo {author} {\bibfnamefont {Pankaj}\ \bibnamefont {Mehta}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Reinforcement {{Learning}} + in {{Different Phases}} of {{Quantum Control}}},}\ }\href {\doibase + 10.1103/PhysRevX.8.031086} {\bibfield {journal} {\bibinfo {journal} + {Physical Review X}\ }\textbf {\bibinfo {volume} {8}},\ \bibinfo {pages} + {031086} (\bibinfo {year} {2018}{\natexlab{b}})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Schau{\ss}}\ \emph {et~al.}(2015)\citenamefont + {Schau{\ss}}, \citenamefont {Zeiher}, \citenamefont {Fukuhara}, \citenamefont + {Hild}, \citenamefont {Cheneau}, \citenamefont {Macr{\`\i}}, \citenamefont + {Pohl}, \citenamefont {Bloch},\ and\ \citenamefont {Gross}}]{Schauss1455}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont + {Schau{\ss}}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Zeiher}}, + \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Fukuhara}}, \bibinfo + {author} {\bibfnamefont {S.}~\bibnamefont {Hild}}, \bibinfo {author} + {\bibfnamefont {M.}~\bibnamefont {Cheneau}}, \bibinfo {author} {\bibfnamefont + {T.}~\bibnamefont {Macr{\`\i}}}, \bibinfo {author} {\bibfnamefont + {T.}~\bibnamefont {Pohl}}, \bibinfo {author} {\bibfnamefont {I.}~\bibnamefont + {Bloch}}, \ and\ \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont + {Gross}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Crystallization + in ising quantum magnets},}\ }\href {\doibase 10.1126/science.1258351} + {\bibfield {journal} {\bibinfo {journal} {Science}\ }\textbf {\bibinfo + {volume} {347}},\ \bibinfo {pages} {1455--1458} (\bibinfo {year} + {2015})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Endres}\ \emph {et~al.}(2016)\citenamefont {Endres}, + \citenamefont {Bernien}, \citenamefont {Keesling}, \citenamefont {Levine}, + \citenamefont {Anschuetz}, \citenamefont {Krajenbrink}, \citenamefont + {Senko}, \citenamefont {Vuletic}, \citenamefont {Greiner},\ and\ + \citenamefont {Lukin}}]{Endres2016}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Manuel}\ \bibnamefont + {Endres}}, \bibinfo {author} {\bibfnamefont {Hannes}\ \bibnamefont + {Bernien}}, \bibinfo {author} {\bibfnamefont {Alexander}\ \bibnamefont + {Keesling}}, \bibinfo {author} {\bibfnamefont {Harry}\ \bibnamefont + {Levine}}, \bibinfo {author} {\bibfnamefont {Eric~R}\ \bibnamefont + {Anschuetz}}, \bibinfo {author} {\bibfnamefont {Alexandre}\ \bibnamefont + {Krajenbrink}}, \bibinfo {author} {\bibfnamefont {Crystal}\ \bibnamefont + {Senko}}, \bibinfo {author} {\bibfnamefont {Vladan}\ \bibnamefont {Vuletic}}, + \bibinfo {author} {\bibfnamefont {Markus}\ \bibnamefont {Greiner}}, \ and\ + \bibinfo {author} {\bibfnamefont {Mikhail~D}\ \bibnamefont {Lukin}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {{Atom-by-atom assembly of + defect-free one-dimensional cold atom arrays}},}\ }\href {\doibase + 10.1126/science.aah3752} {\bibfield {journal} {\bibinfo {journal} + {Science}\ }\textbf {\bibinfo {volume} {354}},\ \bibinfo {pages} {1024--1027} + (\bibinfo {year} {2016})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Labuhn}\ \emph {et~al.}(2016)\citenamefont {Labuhn}, + \citenamefont {Barredo}, \citenamefont {Ravets}, \citenamefont + {de~L{\'e}s{\'e}leuc}, \citenamefont {Macr{\`\i}}, \citenamefont {Lahaye},\ + and\ \citenamefont {Browaeys}}]{Labuhn}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Henning}\ \bibnamefont + {Labuhn}}, \bibinfo {author} {\bibfnamefont {Daniel}\ \bibnamefont + {Barredo}}, \bibinfo {author} {\bibfnamefont {Sylvain}\ \bibnamefont + {Ravets}}, \bibinfo {author} {\bibfnamefont {Sylvain}\ \bibnamefont + {de~L{\'e}s{\'e}leuc}}, \bibinfo {author} {\bibfnamefont {Tommaso}\ + \bibnamefont {Macr{\`\i}}}, \bibinfo {author} {\bibfnamefont {Thierry}\ + \bibnamefont {Lahaye}}, \ and\ \bibinfo {author} {\bibfnamefont {Antoine}\ + \bibnamefont {Browaeys}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Tunable two-dimensional arrays of single rydberg atoms for realizing quantum + ising models},}\ }\href {https://doi.org/10.1038/nature18274} {\bibfield + {journal} {\bibinfo {journal} {Nature}\ }\textbf {\bibinfo {volume} {534}},\ + \bibinfo {pages} {667 EP --} (\bibinfo {year} {2016})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Bernien}\ \emph {et~al.}(2017)\citenamefont + {Bernien}, \citenamefont {Schwartz}, \citenamefont {Keesling}, \citenamefont + {Levine}, \citenamefont {Omran}, \citenamefont {Pichler}, \citenamefont + {Choi}, \citenamefont {Zibrov}, \citenamefont {Endres}, \citenamefont + {Greiner}, \citenamefont {Vuleti{\'{c}}},\ and\ \citenamefont + {Lukin}}]{Bernien2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Hannes}\ \bibnamefont + {Bernien}}, \bibinfo {author} {\bibfnamefont {Sylvain}\ \bibnamefont + {Schwartz}}, \bibinfo {author} {\bibfnamefont {Alexander}\ \bibnamefont + {Keesling}}, \bibinfo {author} {\bibfnamefont {Harry}\ \bibnamefont + {Levine}}, \bibinfo {author} {\bibfnamefont {Ahmed}\ \bibnamefont {Omran}}, + \bibinfo {author} {\bibfnamefont {Hannes}\ \bibnamefont {Pichler}}, \bibinfo + {author} {\bibfnamefont {Soonwon}\ \bibnamefont {Choi}}, \bibinfo {author} + {\bibfnamefont {Alexander~S}\ \bibnamefont {Zibrov}}, \bibinfo {author} + {\bibfnamefont {Manuel}\ \bibnamefont {Endres}}, \bibinfo {author} + {\bibfnamefont {Markus}\ \bibnamefont {Greiner}}, \bibinfo {author} + {\bibfnamefont {Vladan}\ \bibnamefont {Vuleti{\'{c}}}}, \ and\ \bibinfo + {author} {\bibfnamefont {Mikhail~D}\ \bibnamefont {Lukin}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {{Probing many-body dynamics on a 51-atom + quantum simulator}},}\ }\href {\doibase 10.1038/nature24622} {\bibfield + {journal} {\bibinfo {journal} {Nature}\ }\textbf {\bibinfo {volume} {551}},\ + \bibinfo {pages} {579--584} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Keesling}\ \emph {et~al.}(2019)\citenamefont + {Keesling}, \citenamefont {Omran}, \citenamefont {Levine}, \citenamefont + {Bernien}, \citenamefont {Pichler}, \citenamefont {Choi}, \citenamefont + {Samajdar}, \citenamefont {Schwartz}, \citenamefont {Silvi}, \citenamefont + {Sachdev}, \citenamefont {Zoller}, \citenamefont {Endres}, \citenamefont + {Greiner}, \citenamefont {Vuletic},\ and\ \citenamefont + {Lukin}}]{keesling_quantum_2019}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Alexander}\ + \bibnamefont {Keesling}}, \bibinfo {author} {\bibfnamefont {Ahmed}\ + \bibnamefont {Omran}}, \bibinfo {author} {\bibfnamefont {Harry}\ \bibnamefont + {Levine}}, \bibinfo {author} {\bibfnamefont {Hannes}\ \bibnamefont + {Bernien}}, \bibinfo {author} {\bibfnamefont {Hannes}\ \bibnamefont + {Pichler}}, \bibinfo {author} {\bibfnamefont {Soonwon}\ \bibnamefont {Choi}}, + \bibinfo {author} {\bibfnamefont {Rhine}\ \bibnamefont {Samajdar}}, \bibinfo + {author} {\bibfnamefont {Sylvain}\ \bibnamefont {Schwartz}}, \bibinfo + {author} {\bibfnamefont {Pietro}\ \bibnamefont {Silvi}}, \bibinfo {author} + {\bibfnamefont {Subir}\ \bibnamefont {Sachdev}}, \bibinfo {author} + {\bibfnamefont {Peter}\ \bibnamefont {Zoller}}, \bibinfo {author} + {\bibfnamefont {Manuel}\ \bibnamefont {Endres}}, \bibinfo {author} + {\bibfnamefont {Markus}\ \bibnamefont {Greiner}}, \bibinfo {author} + {\bibfnamefont {Vladan}\ \bibnamefont {Vuletic}}, \ and\ \bibinfo {author} + {\bibfnamefont {Mikhail~D.}\ \bibnamefont {Lukin}},\ }\bibfield {title} + {\enquote {\bibinfo {title} {Quantum kibble-zurek mechanism and critical + dynamics on a programmable rydberg simulator},}\ }\href {\doibase + 10.1038/s41586-019-1070-1} {\bibfield {journal} {\bibinfo {journal} + {Nature}\ }\textbf {\bibinfo {volume} {568}},\ \bibinfo {pages} {207--211} + (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Ebadi}}\ \emph {et~al.}(2020)\citenamefont + {{Ebadi}}, \citenamefont {{Wang}}, \citenamefont {{Levine}}, \citenamefont + {{Keesling}}, \citenamefont {{Semeghini}}, \citenamefont {{Omran}}, + \citenamefont {{Bluvstein}}, \citenamefont {{Samajdar}}, \citenamefont + {{Pichler}}, \citenamefont {{Ho}}, \citenamefont {{Choi}}, \citenamefont + {{Sachdev}}, \citenamefont {{Greiner}}, \citenamefont {{Vuletic}},\ and\ + \citenamefont {{Lukin}}}]{2020arXiv201212281E}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Sepehr}\ \bibnamefont + {{Ebadi}}}, \bibinfo {author} {\bibfnamefont {Tout~T.}\ \bibnamefont + {{Wang}}}, \bibinfo {author} {\bibfnamefont {Harry}\ \bibnamefont + {{Levine}}}, \bibinfo {author} {\bibfnamefont {Alexander}\ \bibnamefont + {{Keesling}}}, \bibinfo {author} {\bibfnamefont {Giulia}\ \bibnamefont + {{Semeghini}}}, \bibinfo {author} {\bibfnamefont {Ahmed}\ \bibnamefont + {{Omran}}}, \bibinfo {author} {\bibfnamefont {Dolev}\ \bibnamefont + {{Bluvstein}}}, \bibinfo {author} {\bibfnamefont {Rhine}\ \bibnamefont + {{Samajdar}}}, \bibinfo {author} {\bibfnamefont {Hannes}\ \bibnamefont + {{Pichler}}}, \bibinfo {author} {\bibfnamefont {Wen~Wei}\ \bibnamefont + {{Ho}}}, \bibinfo {author} {\bibfnamefont {Soonwon}\ \bibnamefont {{Choi}}}, + \bibinfo {author} {\bibfnamefont {Subir}\ \bibnamefont {{Sachdev}}}, \bibinfo + {author} {\bibfnamefont {Markus}\ \bibnamefont {{Greiner}}}, \bibinfo + {author} {\bibfnamefont {Vladan}\ \bibnamefont {{Vuletic}}}, \ and\ \bibinfo + {author} {\bibfnamefont {Mikhail~D.}\ \bibnamefont {{Lukin}}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {{Quantum Phases of Matter on a 256-Atom + Programmable Quantum Simulator}},}\ }\href@noop {} {\bibfield {journal} + {\bibinfo {journal} {arXiv e-prints}\ ,\ \bibinfo {eid} {arXiv:2012.12281}} + (\bibinfo {year} {2020})},\ \Eprint {http://arxiv.org/abs/2012.12281} + {arXiv:2012.12281 [quant-ph]} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Scholl}}\ \emph {et~al.}(2020)\citenamefont + {{Scholl}}, \citenamefont {{Schuler}}, \citenamefont {{Williams}}, + \citenamefont {{Eberharter}}, \citenamefont {{Barredo}}, \citenamefont + {{Schymik}}, \citenamefont {{Lienhard}}, \citenamefont {{Henry}}, + \citenamefont {{Lang}}, \citenamefont {{Lahaye}}, \citenamefont + {{L{\"a}uchli}},\ and\ \citenamefont {{Browaeys}}}]{2020arXiv201212268S}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Pascal}\ \bibnamefont + {{Scholl}}}, \bibinfo {author} {\bibfnamefont {Michael}\ \bibnamefont + {{Schuler}}}, \bibinfo {author} {\bibfnamefont {Hannah~J.}\ \bibnamefont + {{Williams}}}, \bibinfo {author} {\bibfnamefont {Alexander~A.}\ \bibnamefont + {{Eberharter}}}, \bibinfo {author} {\bibfnamefont {Daniel}\ \bibnamefont + {{Barredo}}}, \bibinfo {author} {\bibfnamefont {Kai-Niklas}\ \bibnamefont + {{Schymik}}}, \bibinfo {author} {\bibfnamefont {Vincent}\ \bibnamefont + {{Lienhard}}}, \bibinfo {author} {\bibfnamefont {Louis-Paul}\ \bibnamefont + {{Henry}}}, \bibinfo {author} {\bibfnamefont {Thomas~C.}\ \bibnamefont + {{Lang}}}, \bibinfo {author} {\bibfnamefont {Thierry}\ \bibnamefont + {{Lahaye}}}, \bibinfo {author} {\bibfnamefont {Andreas~M.}\ \bibnamefont + {{L{\"a}uchli}}}, \ and\ \bibinfo {author} {\bibfnamefont {Antoine}\ + \bibnamefont {{Browaeys}}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {{Programmable quantum simulation of 2D antiferromagnets with hundreds of + Rydberg atoms}},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} + {arXiv e-prints}\ ,\ \bibinfo {eid} {arXiv:2012.12268}} (\bibinfo {year} + {2020})},\ \Eprint {http://arxiv.org/abs/2012.12268} {arXiv:2012.12268 + [quant-ph]} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Levine}\ \emph {et~al.}(2019)\citenamefont {Levine}, + \citenamefont {Keesling}, \citenamefont {Semeghini}, \citenamefont {Omran}, + \citenamefont {Wang}, \citenamefont {Ebadi}, \citenamefont {Bernien}, + \citenamefont {Greiner}, \citenamefont {Vuleti\ifmmode~\acute{c}\else + \'{c}\fi{}}, \citenamefont {Pichler},\ and\ \citenamefont + {Lukin}}]{Levine2019}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Harry}\ \bibnamefont + {Levine}}, \bibinfo {author} {\bibfnamefont {Alexander}\ \bibnamefont + {Keesling}}, \bibinfo {author} {\bibfnamefont {Giulia}\ \bibnamefont + {Semeghini}}, \bibinfo {author} {\bibfnamefont {Ahmed}\ \bibnamefont + {Omran}}, \bibinfo {author} {\bibfnamefont {Tout~T.}\ \bibnamefont {Wang}}, + \bibinfo {author} {\bibfnamefont {Sepehr}\ \bibnamefont {Ebadi}}, \bibinfo + {author} {\bibfnamefont {Hannes}\ \bibnamefont {Bernien}}, \bibinfo {author} + {\bibfnamefont {Markus}\ \bibnamefont {Greiner}}, \bibinfo {author} + {\bibfnamefont {Vladan}\ \bibnamefont {Vuleti\ifmmode~\acute{c}\else + \'{c}\fi{}}}, \bibinfo {author} {\bibfnamefont {Hannes}\ \bibnamefont + {Pichler}}, \ and\ \bibinfo {author} {\bibfnamefont {Mikhail~D.}\ + \bibnamefont {Lukin}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Parallel implementation of high-fidelity multiqubit gates with neutral + atoms},}\ }\href {\doibase 10.1103/PhysRevLett.123.170503} {\bibfield + {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo + {volume} {123}},\ \bibinfo {pages} {170503} (\bibinfo {year} + {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Samajdar}\ \emph {et~al.}(2020)\citenamefont + {Samajdar}, \citenamefont {Ho}, \citenamefont {Pichler}, \citenamefont + {Lukin},\ and\ \citenamefont {Sachdev}}]{Samajdar2020}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Rhine}\ \bibnamefont + {Samajdar}}, \bibinfo {author} {\bibfnamefont {Wen~Wei}\ \bibnamefont {Ho}}, + \bibinfo {author} {\bibfnamefont {Hannes}\ \bibnamefont {Pichler}}, \bibinfo + {author} {\bibfnamefont {Mikhail~D.}\ \bibnamefont {Lukin}}, \ and\ \bibinfo + {author} {\bibfnamefont {Subir}\ \bibnamefont {Sachdev}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Complex density wave orders and quantum + phase transitions in a model of square-lattice rydberg atom arrays},}\ }\href + {\doibase 10.1103/PhysRevLett.124.103601} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {124}},\ \bibinfo + {pages} {103601} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {White}(1992)}]{PhysRevLett.69.2863}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Steven~R.}\ + \bibnamefont {White}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Density matrix formulation for quantum renormalization groups},}\ }\href + {\doibase 10.1103/PhysRevLett.69.2863} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {69}},\ \bibinfo + {pages} {2863--2866} (\bibinfo {year} {1992})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {White}(1993)}]{PhysRevB.48.10345}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Steven~R.}\ + \bibnamefont {White}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Density-matrix algorithms for quantum renormalization groups},}\ }\href + {\doibase 10.1103/PhysRevB.48.10345} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {48}},\ \bibinfo + {pages} {10345--10356} (\bibinfo {year} {1993})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Schollwoeck}(2011)}]{SCHOLLWOCK201196}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Ulrich}\ \bibnamefont + {Schollwoeck}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The + density-matrix renormalization group in the age of matrix product states},}\ + }\href {\doibase https://doi.org/10.1016/j.aop.2010.09.012} {\bibfield + {journal} {\bibinfo {journal} {Annals of Physics}\ }\textbf {\bibinfo + {volume} {326}},\ \bibinfo {pages} {96 -- 192} (\bibinfo {year} {2011})},\ + \bibinfo {note} {january 2011 Special Issue}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Fishman}}\ \emph {et~al.}(2020)\citenamefont + {{Fishman}}, \citenamefont {{White}},\ and\ \citenamefont {{Miles + Stoudenmire}}}]{itensor}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Matthew}\ \bibnamefont + {{Fishman}}}, \bibinfo {author} {\bibfnamefont {Steven~R.}\ \bibnamefont + {{White}}}, \ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {{Miles + Stoudenmire}}},\ }\bibfield {title} {\enquote {\bibinfo {title} {{The + ITensor Software Library for Tensor Network Calculations}},}\ }\href@noop {} + {\bibfield {journal} {\bibinfo {journal} {arXiv e-prints}\ ,\ \bibinfo + {eid} {arXiv:2007.14822}} (\bibinfo {year} {2020})},\ \Eprint + {http://arxiv.org/abs/2007.14822} {arXiv:2007.14822 [cs.MS]} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Ferris}\ and\ \citenamefont + {Vidal}(2012)}]{PhysRevB.85.165146}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Andrew~J.}\ + \bibnamefont {Ferris}}\ and\ \bibinfo {author} {\bibfnamefont {Guifre}\ + \bibnamefont {Vidal}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Perfect sampling with unitary tensor networks},}\ }\href {\doibase + 10.1103/PhysRevB.85.165146} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. B}\ }\textbf {\bibinfo {volume} {85}},\ \bibinfo {pages} {165146} + (\bibinfo {year} {2012})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Basko}\ \emph {et~al.}(2006)\citenamefont {Basko}, + \citenamefont {Aleiner},\ and\ \citenamefont {Altshuler}}]{basko2006}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~M.}\ \bibnamefont + {Basko}}, \bibinfo {author} {\bibfnamefont {I.~L.}\ \bibnamefont {Aleiner}}, + \ and\ \bibinfo {author} {\bibfnamefont {B.~L.}\ \bibnamefont {Altshuler}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Metal\textendash insulator + transition in a weakly interacting many-electron system with localized + single-particle states},}\ }\href {\doibase 10.1016/j.aop.2005.11.014} + {\bibfield {journal} {\bibinfo {journal} {Annals of Physics}\ }\textbf + {\bibinfo {volume} {321}},\ \bibinfo {pages} {1126--1205} (\bibinfo {year} + {2006})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Abanin}\ \emph {et~al.}(2019)\citenamefont {Abanin}, + \citenamefont {Altman}, \citenamefont {Bloch},\ and\ \citenamefont + {Serbyn}}]{RevModPhys.91.021001}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Dmitry~A.}\ + \bibnamefont {Abanin}}, \bibinfo {author} {\bibfnamefont {Ehud}\ \bibnamefont + {Altman}}, \bibinfo {author} {\bibfnamefont {Immanuel}\ \bibnamefont + {Bloch}}, \ and\ \bibinfo {author} {\bibfnamefont {Maksym}\ \bibnamefont + {Serbyn}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Colloquium: + Many-body localization, thermalization, and entanglement},}\ }\href {\doibase + 10.1103/RevModPhys.91.021001} {\bibfield {journal} {\bibinfo {journal} + {Rev. Mod. Phys.}\ }\textbf {\bibinfo {volume} {91}},\ \bibinfo {pages} + {021001} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Savary}\ and\ \citenamefont + {Balents}(2016)}]{savaryQuantumSpinLiquids2016}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Lucile}\ \bibnamefont + {Savary}}\ and\ \bibinfo {author} {\bibfnamefont {Leon}\ \bibnamefont + {Balents}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum spin + liquids: A review},}\ }\href {\doibase 10.1088/0034-4885/80/1/016502} + {\bibfield {journal} {\bibinfo {journal} {Reports on Progress in Physics}\ + }\textbf {\bibinfo {volume} {80}},\ \bibinfo {pages} {016502} (\bibinfo + {year} {2016})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Knolle}\ and\ \citenamefont + {Moessner}(2019)}]{doi:10.1146/annurev-conmatphys-031218-013401}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont + {Knolle}}\ and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont + {Moessner}},\ }\bibfield {title} {\enquote {\bibinfo {title} {A field guide + to spin liquids},}\ }\href {\doibase + 10.1146/annurev-conmatphys-031218-013401} {\bibfield {journal} {\bibinfo + {journal} {Annual Review of Condensed Matter Physics}\ }\textbf {\bibinfo + {volume} {10}},\ \bibinfo {pages} {451--472} (\bibinfo {year} {2019})},\ + \Eprint + {http://arxiv.org/abs/https://doi.org/10.1146/annurev-conmatphys-031218-013401} + {https://doi.org/10.1146/annurev-conmatphys-031218-013401} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Broecker}\ \emph + {et~al.}(2017{\natexlab{b}})\citenamefont {Broecker}, \citenamefont + {Assaad},\ and\ \citenamefont {Trebst}}]{broecker2017b}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Peter}\ \bibnamefont + {Broecker}}, \bibinfo {author} {\bibfnamefont {Fakher~F.}\ \bibnamefont + {Assaad}}, \ and\ \bibinfo {author} {\bibfnamefont {Simon}\ \bibnamefont + {Trebst}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum phase + recognition via unsupervised machine learning},}\ }\href@noop {} {\bibfield + {journal} {\bibinfo {journal} {arXiv:1707.00663 [cond-mat]}\ } (\bibinfo + {year} {2017}{\natexlab{b}})},\ \Eprint {http://arxiv.org/abs/1707.00663} + {arXiv:1707.00663 [cond-mat]} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Wetzel}(2017)}]{PhysRevE.96.022140}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Sebastian~J.}\ + \bibnamefont {Wetzel}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Unsupervised learning of phase transitions: From principal component + analysis to variational autoencoders},}\ }\href {\doibase + 10.1103/PhysRevE.96.022140} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. E}\ }\textbf {\bibinfo {volume} {96}},\ \bibinfo {pages} {022140} + (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Rodriguez-Nieva}}\ and\ \citenamefont + {Scheurer}(2019)}]{rodriguez-nieva2019}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Joaquin~F.}\ + \bibnamefont {{Rodriguez-Nieva}}}\ and\ \bibinfo {author} {\bibfnamefont + {Mathias~S.}\ \bibnamefont {Scheurer}},\ }\bibfield {title} {\enquote + {\bibinfo {title} {Identifying topological order through unsupervised machine + learning},}\ }\href {\doibase 10.1038/s41567-019-0512-x} {\bibfield + {journal} {\bibinfo {journal} {Nature Physics}\ }\textbf {\bibinfo {volume} + {15}},\ \bibinfo {pages} {790--795} (\bibinfo {year} {2019})}\BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Lustig}\ \emph {et~al.}(2020)\citenamefont {Lustig}, + \citenamefont {Yair}, \citenamefont {Talmon},\ and\ \citenamefont + {Segev}}]{PhysRevLett.125.127401}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Eran}\ \bibnamefont + {Lustig}}, \bibinfo {author} {\bibfnamefont {Or}~\bibnamefont {Yair}}, + \bibinfo {author} {\bibfnamefont {Ronen}\ \bibnamefont {Talmon}}, \ and\ + \bibinfo {author} {\bibfnamefont {Mordechai}\ \bibnamefont {Segev}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Identifying topological + phase transitions in experiments using manifold learning},}\ }\href {\doibase + 10.1103/PhysRevLett.125.127401} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {125}},\ \bibinfo {pages} + {127401} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Rao}(2018)}]{rao2018}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Wen-Jia}\ \bibnamefont + {Rao}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Machine learning + the many-body localization transition in random spin systems},}\ }\href + {\doibase 10.1088/1361-648X/aaddc6} {\bibfield {journal} {\bibinfo + {journal} {Journal of Physics: Condensed Matter}\ }\textbf {\bibinfo {volume} + {30}},\ \bibinfo {pages} {395902} (\bibinfo {year} {2018})}\BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Venderley}\ \emph {et~al.}(2018)\citenamefont + {Venderley}, \citenamefont {Khemani},\ and\ \citenamefont + {Kim}}]{PhysRevLett.120.257204}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Jordan}\ \bibnamefont + {Venderley}}, \bibinfo {author} {\bibfnamefont {Vedika}\ \bibnamefont + {Khemani}}, \ and\ \bibinfo {author} {\bibfnamefont {Eun-Ah}\ \bibnamefont + {Kim}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Machine learning + out-of-equilibrium phases of matter},}\ }\href {\doibase + 10.1103/PhysRevLett.120.257204} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {120}},\ \bibinfo {pages} + {257204} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Dong}\ \emph {et~al.}(2019)\citenamefont {Dong}, + \citenamefont {Pollmann},\ and\ \citenamefont {Zhang}}]{PhysRevB.99.121104}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Xiao-Yu}\ \bibnamefont + {Dong}}, \bibinfo {author} {\bibfnamefont {Frank}\ \bibnamefont {Pollmann}}, + \ and\ \bibinfo {author} {\bibfnamefont {Xue-Feng}\ \bibnamefont {Zhang}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Machine learning of quantum + phase transitions},}\ }\href {\doibase 10.1103/PhysRevB.99.121104} {\bibfield + {journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo {volume} + {99}},\ \bibinfo {pages} {121104} (\bibinfo {year} {2019})}\BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Tsai}\ \emph {et~al.}(2020)\citenamefont {Tsai}, + \citenamefont {Yu}, \citenamefont {Hsu},\ and\ \citenamefont + {Chung}}]{PhysRevB.102.054512}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Yuan-Hong}\ + \bibnamefont {Tsai}}, \bibinfo {author} {\bibfnamefont {Meng-Zhe}\ + \bibnamefont {Yu}}, \bibinfo {author} {\bibfnamefont {Yu-Hao}\ \bibnamefont + {Hsu}}, \ and\ \bibinfo {author} {\bibfnamefont {Ming-Chiang}\ \bibnamefont + {Chung}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Deep learning of + topological phase transitions from entanglement aspects},}\ }\href {\doibase + 10.1103/PhysRevB.102.054512} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. B}\ }\textbf {\bibinfo {volume} {102}},\ \bibinfo {pages} + {054512} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Kottmann}\ \emph {et~al.}(2020)\citenamefont + {Kottmann}, \citenamefont {Huembeli}, \citenamefont {Lewenstein},\ and\ + \citenamefont {Ac\'{\i}n}}]{PhysRevLett.125.170603}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Korbinian}\ + \bibnamefont {Kottmann}}, \bibinfo {author} {\bibfnamefont {Patrick}\ + \bibnamefont {Huembeli}}, \bibinfo {author} {\bibfnamefont {Maciej}\ + \bibnamefont {Lewenstein}}, \ and\ \bibinfo {author} {\bibfnamefont + {Antonio}\ \bibnamefont {Ac\'{\i}n}},\ }\bibfield {title} {\enquote + {\bibinfo {title} {Unsupervised phase discovery with deep anomaly + detection},}\ }\href {\doibase 10.1103/PhysRevLett.125.170603} {\bibfield + {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo + {volume} {125}},\ \bibinfo {pages} {170603} (\bibinfo {year} + {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Berezutskii}\ \emph {et~al.}(2020)\citenamefont + {Berezutskii}, \citenamefont {Beketov}, \citenamefont {Yudin}, \citenamefont + {Zimbor{\'a}s},\ and\ \citenamefont {Biamonte}}]{berezutskii2020}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont + {Berezutskii}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont + {Beketov}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Yudin}}, + \bibinfo {author} {\bibfnamefont {Z.}~\bibnamefont {Zimbor{\'a}s}}, \ and\ + \bibinfo {author} {\bibfnamefont {J.~D.}\ \bibnamefont {Biamonte}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Probing criticality in + quantum spin chains with neural networks},}\ }\href {\doibase + 10.1088/2632-072X/abaa2b} {\bibfield {journal} {\bibinfo {journal} {Journal + of Physics: Complexity}\ }\textbf {\bibinfo {volume} {1}},\ \bibinfo {pages} + {03LT01} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Khatami}\ \emph {et~al.}(2020)\citenamefont + {Khatami}, \citenamefont {Guardado-Sanchez}, \citenamefont {Spar}, + \citenamefont {Carrasquilla}, \citenamefont {Bakr},\ and\ \citenamefont + {Scalettar}}]{PhysRevA.102.033326}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Ehsan}\ \bibnamefont + {Khatami}}, \bibinfo {author} {\bibfnamefont {Elmer}\ \bibnamefont + {Guardado-Sanchez}}, \bibinfo {author} {\bibfnamefont {Benjamin~M.}\ + \bibnamefont {Spar}}, \bibinfo {author} {\bibfnamefont {Juan~Felipe}\ + \bibnamefont {Carrasquilla}}, \bibinfo {author} {\bibfnamefont {Waseem~S.}\ + \bibnamefont {Bakr}}, \ and\ \bibinfo {author} {\bibfnamefont {Richard~T.}\ + \bibnamefont {Scalettar}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Visualizing strange metallic correlations in the two-dimensional + fermi-hubbard model with artificial intelligence},}\ }\href {\doibase + 10.1103/PhysRevA.102.033326} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {102}},\ \bibinfo {pages} + {033326} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Baydin}\ \emph {et~al.}(2017)\citenamefont {Baydin}, + \citenamefont {Pearlmutter}, \citenamefont {Radul},\ and\ \citenamefont + {Siskind}}]{AD_2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Atilim~Gunes}\ + \bibnamefont {Baydin}}, \bibinfo {author} {\bibfnamefont {Barak~A.}\ + \bibnamefont {Pearlmutter}}, \bibinfo {author} {\bibfnamefont + {Alexey~Andreyevich}\ \bibnamefont {Radul}}, \ and\ \bibinfo {author} + {\bibfnamefont {Jeffrey~Mark}\ \bibnamefont {Siskind}},\ }\bibfield {title} + {\enquote {\bibinfo {title} {Automatic differentiation in machine learning: A + survey},}\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {J. + Mach. Learn. Res.}\ }\textbf {\bibinfo {volume} {18}} (\bibinfo {year} + {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Kingma}}\ and\ \citenamefont + {{Ba}}(2014)}]{2014arXiv1412.6980K}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Diederik~P.}\ + \bibnamefont {{Kingma}}}\ and\ \bibinfo {author} {\bibfnamefont {Jimmy}\ + \bibnamefont {{Ba}}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {{Adam: A Method for Stochastic Optimization}},}\ }\href@noop {} {\bibfield + {journal} {\bibinfo {journal} {arXiv e-prints}\ ,\ \bibinfo {eid} + {arXiv:1412.6980}} (\bibinfo {year} {2014})},\ \Eprint + {http://arxiv.org/abs/1412.6980} {arXiv:1412.6980 [cs.LG]} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Abadi}\ \emph {et~al.}(2015)\citenamefont {Abadi}, + \citenamefont {Agarwal}, \citenamefont {Barham}, \citenamefont {Brevdo}, + \citenamefont {Chen}, \citenamefont {Citro}, \citenamefont {Corrado}, + \citenamefont {Davis}, \citenamefont {Dean}, \citenamefont {Devin}, + \citenamefont {Ghemawat}, \citenamefont {Goodfellow}, \citenamefont {Harp}, + \citenamefont {Irving}, \citenamefont {Isard}, \citenamefont {Jia}, + \citenamefont {Jozefowicz}, \citenamefont {Kaiser}, \citenamefont {Kudlur}, + \citenamefont {Levenberg}, \citenamefont {Man\'{e}}, \citenamefont {Monga}, + \citenamefont {Moore}, \citenamefont {Murray}, \citenamefont {Olah}, + \citenamefont {Schuster}, \citenamefont {Shlens}, \citenamefont {Steiner}, + \citenamefont {Sutskever}, \citenamefont {Talwar}, \citenamefont {Tucker}, + \citenamefont {Vanhoucke}, \citenamefont {Vasudevan}, \citenamefont + {Vi\'{e}gas}, \citenamefont {Vinyals}, \citenamefont {Warden}, \citenamefont + {Wattenberg}, \citenamefont {Wicke}, \citenamefont {Yu},\ and\ \citenamefont + {Zheng}}]{tensorflow}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Mart\'{\i}n}\ + \bibnamefont {Abadi}}, \bibinfo {author} {\bibfnamefont {Ashish}\ + \bibnamefont {Agarwal}}, \bibinfo {author} {\bibfnamefont {Paul}\ + \bibnamefont {Barham}}, \bibinfo {author} {\bibfnamefont {Eugene}\ + \bibnamefont {Brevdo}}, \bibinfo {author} {\bibfnamefont {Zhifeng}\ + \bibnamefont {Chen}}, \bibinfo {author} {\bibfnamefont {Craig}\ \bibnamefont + {Citro}}, \bibinfo {author} {\bibfnamefont {Greg~S.}\ \bibnamefont + {Corrado}}, \bibinfo {author} {\bibfnamefont {Andy}\ \bibnamefont {Davis}}, + \bibinfo {author} {\bibfnamefont {Jeffrey}\ \bibnamefont {Dean}}, \bibinfo + {author} {\bibfnamefont {Matthieu}\ \bibnamefont {Devin}}, \bibinfo {author} + {\bibfnamefont {Sanjay}\ \bibnamefont {Ghemawat}}, \bibinfo {author} + {\bibfnamefont {Ian}\ \bibnamefont {Goodfellow}}, \bibinfo {author} + {\bibfnamefont {Andrew}\ \bibnamefont {Harp}}, \bibinfo {author} + {\bibfnamefont {Geoffrey}\ \bibnamefont {Irving}}, \bibinfo {author} + {\bibfnamefont {Michael}\ \bibnamefont {Isard}}, \bibinfo {author} + {\bibfnamefont {Yangqing}\ \bibnamefont {Jia}}, \bibinfo {author} + {\bibfnamefont {Rafal}\ \bibnamefont {Jozefowicz}}, \bibinfo {author} + {\bibfnamefont {Lukasz}\ \bibnamefont {Kaiser}}, \bibinfo {author} + {\bibfnamefont {Manjunath}\ \bibnamefont {Kudlur}}, \bibinfo {author} + {\bibfnamefont {Josh}\ \bibnamefont {Levenberg}}, \bibinfo {author} + {\bibfnamefont {Dan}\ \bibnamefont {Man\'{e}}}, \bibinfo {author} + {\bibfnamefont {Rajat}\ \bibnamefont {Monga}}, \bibinfo {author} + {\bibfnamefont {Sherry}\ \bibnamefont {Moore}}, \bibinfo {author} + {\bibfnamefont {Derek}\ \bibnamefont {Murray}}, \bibinfo {author} + {\bibfnamefont {Chris}\ \bibnamefont {Olah}}, \bibinfo {author} + {\bibfnamefont {Mike}\ \bibnamefont {Schuster}}, \bibinfo {author} + {\bibfnamefont {Jonathon}\ \bibnamefont {Shlens}}, \bibinfo {author} + {\bibfnamefont {Benoit}\ \bibnamefont {Steiner}}, \bibinfo {author} + {\bibfnamefont {Ilya}\ \bibnamefont {Sutskever}}, \bibinfo {author} + {\bibfnamefont {Kunal}\ \bibnamefont {Talwar}}, \bibinfo {author} + {\bibfnamefont {Paul}\ \bibnamefont {Tucker}}, \bibinfo {author} + {\bibfnamefont {Vincent}\ \bibnamefont {Vanhoucke}}, \bibinfo {author} + {\bibfnamefont {Vijay}\ \bibnamefont {Vasudevan}}, \bibinfo {author} + {\bibfnamefont {Fernanda}\ \bibnamefont {Vi\'{e}gas}}, \bibinfo {author} + {\bibfnamefont {Oriol}\ \bibnamefont {Vinyals}}, \bibinfo {author} + {\bibfnamefont {Pete}\ \bibnamefont {Warden}}, \bibinfo {author} + {\bibfnamefont {Martin}\ \bibnamefont {Wattenberg}}, \bibinfo {author} + {\bibfnamefont {Martin}\ \bibnamefont {Wicke}}, \bibinfo {author} + {\bibfnamefont {Yuan}\ \bibnamefont {Yu}}, \ and\ \bibinfo {author} + {\bibfnamefont {Xiaoqiang}\ \bibnamefont {Zheng}},\ }\href + {http://tensorflow.org/} {\enquote {\bibinfo {title} {{TensorFlow}: + Large-scale machine learning on heterogeneous systems},}\ } (\bibinfo {year} + {2015}),\ \bibinfo {note} {software available from + tensorflow.org}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Eisert}\ \emph {et~al.}(2020)\citenamefont {Eisert}, + \citenamefont {Hangleiter}, \citenamefont {Walk}, \citenamefont {Roth}, + \citenamefont {Markham}, \citenamefont {Parekh}, \citenamefont {Chabaud},\ + and\ \citenamefont {Kashefi}}]{eisert_quantum_2020}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Jens}\ \bibnamefont + {Eisert}}, \bibinfo {author} {\bibfnamefont {Dominik}\ \bibnamefont + {Hangleiter}}, \bibinfo {author} {\bibfnamefont {Nathan}\ \bibnamefont + {Walk}}, \bibinfo {author} {\bibfnamefont {Ingo}\ \bibnamefont {Roth}}, + \bibinfo {author} {\bibfnamefont {Damian}\ \bibnamefont {Markham}}, \bibinfo + {author} {\bibfnamefont {Rhea}\ \bibnamefont {Parekh}}, \bibinfo {author} + {\bibfnamefont {Ulysse}\ \bibnamefont {Chabaud}}, \ and\ \bibinfo {author} + {\bibfnamefont {Elham}\ \bibnamefont {Kashefi}},\ }\bibfield {title} + {\enquote {\bibinfo {title} {Quantum certification and benchmarking},}\ + }\href {\doibase 10.1038/s42254-020-0186-4} {\bibfield {journal} {\bibinfo + {journal} {Nature Reviews Physics}\ }\textbf {\bibinfo {volume} {2}},\ + \bibinfo {pages} {382--390} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Knill}\ \emph {et~al.}(2008)\citenamefont {Knill}, + \citenamefont {Leibfried}, \citenamefont {Reichle}, \citenamefont {Britton}, + \citenamefont {Blakestad}, \citenamefont {Jost}, \citenamefont {Langer}, + \citenamefont {Ozeri}, \citenamefont {Seidelin},\ and\ \citenamefont + {Wineland}}]{PhysRevA.77.012307}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont + {Knill}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Leibfried}}, + \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Reichle}}, \bibinfo + {author} {\bibfnamefont {J.}~\bibnamefont {Britton}}, \bibinfo {author} + {\bibfnamefont {R.~B.}\ \bibnamefont {Blakestad}}, \bibinfo {author} + {\bibfnamefont {J.~D.}\ \bibnamefont {Jost}}, \bibinfo {author} + {\bibfnamefont {C.}~\bibnamefont {Langer}}, \bibinfo {author} {\bibfnamefont + {R.}~\bibnamefont {Ozeri}}, \bibinfo {author} {\bibfnamefont + {S.}~\bibnamefont {Seidelin}}, \ and\ \bibinfo {author} {\bibfnamefont + {D.~J.}\ \bibnamefont {Wineland}},\ }\bibfield {title} {\enquote {\bibinfo + {title} {Randomized benchmarking of quantum gates},}\ }\href {\doibase + 10.1103/PhysRevA.77.012307} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. A}\ }\textbf {\bibinfo {volume} {77}},\ \bibinfo {pages} {012307} + (\bibinfo {year} {2008})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Magesan}\ \emph {et~al.}(2012)\citenamefont + {Magesan}, \citenamefont {Gambetta},\ and\ \citenamefont + {Emerson}}]{PhysRevA.85.042311}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Easwar}\ \bibnamefont + {Magesan}}, \bibinfo {author} {\bibfnamefont {Jay~M.}\ \bibnamefont + {Gambetta}}, \ and\ \bibinfo {author} {\bibfnamefont {Joseph}\ \bibnamefont + {Emerson}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Characterizing + quantum gates via randomized benchmarking},}\ }\href {\doibase + 10.1103/PhysRevA.85.042311} {\bibfield {journal} {\bibinfo {journal} {Phys. + Rev. A}\ }\textbf {\bibinfo {volume} {85}},\ \bibinfo {pages} {042311} + (\bibinfo {year} {2012})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Harper}\ \emph {et~al.}(2019)\citenamefont {Harper}, + \citenamefont {Hincks}, \citenamefont {Ferrie}, \citenamefont {Flammia},\ + and\ \citenamefont {Wallman}}]{PhysRevA.99.052350}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Robin}\ \bibnamefont + {Harper}}, \bibinfo {author} {\bibfnamefont {Ian}\ \bibnamefont {Hincks}}, + \bibinfo {author} {\bibfnamefont {Chris}\ \bibnamefont {Ferrie}}, \bibinfo + {author} {\bibfnamefont {Steven~T.}\ \bibnamefont {Flammia}}, \ and\ \bibinfo + {author} {\bibfnamefont {Joel~J.}\ \bibnamefont {Wallman}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Statistical analysis of randomized + benchmarking},}\ }\href {\doibase 10.1103/PhysRevA.99.052350} {\bibfield + {journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} + {99}},\ \bibinfo {pages} {052350} (\bibinfo {year} {2019})}\BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Flammia}\ and\ \citenamefont + {Liu}(2011)}]{PhysRevLett.106.230501}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Steven~T.}\ + \bibnamefont {Flammia}}\ and\ \bibinfo {author} {\bibfnamefont {Yi-Kai}\ + \bibnamefont {Liu}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Direct + fidelity estimation from few pauli measurements},}\ }\href {\doibase + 10.1103/PhysRevLett.106.230501} {\bibfield {journal} {\bibinfo {journal} + {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {106}},\ \bibinfo {pages} + {230501} (\bibinfo {year} {2011})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Aolita}\ \emph {et~al.}(2015)\citenamefont {Aolita}, + \citenamefont {Gogolin}, \citenamefont {Kliesch},\ and\ \citenamefont + {Eisert}}]{aolita_reliable_2015}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Leandro}\ \bibnamefont + {Aolita}}, \bibinfo {author} {\bibfnamefont {Christian}\ \bibnamefont + {Gogolin}}, \bibinfo {author} {\bibfnamefont {Martin}\ \bibnamefont + {Kliesch}}, \ and\ \bibinfo {author} {\bibfnamefont {Jens}\ \bibnamefont + {Eisert}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Reliable quantum + certification of photonic state preparations},}\ }\href {\doibase + 10.1038/ncomms9498} {\bibfield {journal} {\bibinfo {journal} {Nature + Communications}\ }\textbf {\bibinfo {volume} {6}},\ \bibinfo {pages} {8498} + (\bibinfo {year} {2015})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Gluza}\ \emph {et~al.}(2018)\citenamefont {Gluza}, + \citenamefont {Kliesch}, \citenamefont {Eisert},\ and\ \citenamefont + {Aolita}}]{PhysRevLett.120.190501}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont + {Gluza}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Kliesch}}, + \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Eisert}}, \ and\ \bibinfo + {author} {\bibfnamefont {L.}~\bibnamefont {Aolita}},\ }\bibfield {title} + {\enquote {\bibinfo {title} {Fidelity witnesses for fermionic quantum + simulations},}\ }\href {\doibase 10.1103/PhysRevLett.120.190501} {\bibfield + {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo + {volume} {120}},\ \bibinfo {pages} {190501} (\bibinfo {year} + {2018})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Vogel}\ and\ \citenamefont {Risken}(1989)}]{vogel89}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {K.}~\bibnamefont + {Vogel}}\ and\ \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Risken}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {Determination of + quasiprobability distributions in terms of probability distributions for the + rotated quadrature phase},}\ }\href {\doibase 10.1103/PhysRevA.40.2847} + {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo + {volume} {40}},\ \bibinfo {pages} {2847--2849} (\bibinfo {year} + {1989})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Hradil}(1997)}]{PhysRevA.55.R1561}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Z.}~\bibnamefont + {Hradil}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Quantum-state + estimation},}\ }\href {\doibase 10.1103/PhysRevA.55.R1561} {\bibfield + {journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} + {55}},\ \bibinfo {pages} {R1561--R1564} (\bibinfo {year} {1997})}\BibitemShut + {NoStop}% +\bibitem [{\citenamefont {\ifmmode \check{R}\else + \v{R}\fi{}eh\'a\ifmmode~\check{c}\else \v{c}\fi{}ek}\ \emph + {et~al.}(2001)\citenamefont {\ifmmode \check{R}\else + \v{R}\fi{}eh\'a\ifmmode~\check{c}\else \v{c}\fi{}ek}, \citenamefont + {Hradil},\ and\ \citenamefont {Je\ifmmode~\check{z}\else + \v{z}\fi{}ek}}]{PhysRevA.63.040303}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont + {\ifmmode \check{R}\else \v{R}\fi{}eh\'a\ifmmode~\check{c}\else + \v{c}\fi{}ek}}, \bibinfo {author} {\bibfnamefont {Z.}~\bibnamefont {Hradil}}, + \ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont + {Je\ifmmode~\check{z}\else \v{z}\fi{}ek}},\ }\bibfield {title} {\enquote + {\bibinfo {title} {Iterative algorithm for reconstruction of entangled + states},}\ }\href {\doibase 10.1103/PhysRevA.63.040303} {\bibfield {journal} + {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {63}},\ + \bibinfo {pages} {040303} (\bibinfo {year} {2001})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {James}\ \emph {et~al.}(2001)\citenamefont {James}, + \citenamefont {Kwiat}, \citenamefont {Munro},\ and\ \citenamefont + {White}}]{PhysRevA.64.052312}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Daniel F.~V.}\ + \bibnamefont {James}}, \bibinfo {author} {\bibfnamefont {Paul~G.}\ + \bibnamefont {Kwiat}}, \bibinfo {author} {\bibfnamefont {William~J.}\ + \bibnamefont {Munro}}, \ and\ \bibinfo {author} {\bibfnamefont {Andrew~G.}\ + \bibnamefont {White}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Measurement of qubits},}\ }\href {\doibase 10.1103/PhysRevA.64.052312} + {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. A}\ }\textbf {\bibinfo + {volume} {64}},\ \bibinfo {pages} {052312} (\bibinfo {year} + {2001})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Je{\v{z}}ek}\ \emph {et~al.}(2003)\citenamefont + {Je{\v{z}}ek}, \citenamefont {Fiur{\'{a}}{\v{s}}ek},\ and\ \citenamefont + {Hradil}}]{Jezek2003}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Miroslav}\ + \bibnamefont {Je{\v{z}}ek}}, \bibinfo {author} {\bibfnamefont + {Jarom{\'{i}}r}\ \bibnamefont {Fiur{\'{a}}{\v{s}}ek}}, \ and\ \bibinfo + {author} {\bibfnamefont {Zden{\v{e}}k}\ \bibnamefont {Hradil}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {{Quantum inference of states and + processes}},}\ }\href {\doibase 10.1103/PhysRevA.68.012305} {\bibfield + {journal} {\bibinfo {journal} {Physical Review A}\ }\textbf {\bibinfo + {volume} {68}},\ \bibinfo {pages} {012305} (\bibinfo {year} + {2003})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Blume-Kohout}(2010)}]{Blume_Kohout_2010}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Robin}\ \bibnamefont + {Blume-Kohout}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Optimal, + reliable estimation of quantum states},}\ }\href {\doibase + 10.1088/1367-2630/12/4/043034} {\bibfield {journal} {\bibinfo {journal} + {New Journal of Physics}\ }\textbf {\bibinfo {volume} {12}},\ \bibinfo + {pages} {043034} (\bibinfo {year} {2010})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Smolin}\ \emph {et~al.}(2012)\citenamefont {Smolin}, + \citenamefont {Gambetta},\ and\ \citenamefont + {Smith}}]{PhysRevLett.108.070502}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {John~A.}\ \bibnamefont + {Smolin}}, \bibinfo {author} {\bibfnamefont {Jay~M.}\ \bibnamefont + {Gambetta}}, \ and\ \bibinfo {author} {\bibfnamefont {Graeme}\ \bibnamefont + {Smith}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Efficient method + for computing the maximum-likelihood quantum state from measurements with + additive gaussian noise},}\ }\href {\doibase 10.1103/PhysRevLett.108.070502} + {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf + {\bibinfo {volume} {108}},\ \bibinfo {pages} {070502} (\bibinfo {year} + {2012})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Granade}\ \emph {et~al.}(2017)\citenamefont + {Granade}, \citenamefont {Ferrie},\ and\ \citenamefont + {Flammia}}]{Granade_2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Christopher}\ + \bibnamefont {Granade}}, \bibinfo {author} {\bibfnamefont {Christopher}\ + \bibnamefont {Ferrie}}, \ and\ \bibinfo {author} {\bibfnamefont {Steven~T}\ + \bibnamefont {Flammia}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Practical adaptive quantum tomography},}\ }\href {\doibase + 10.1088/1367-2630/aa8fe6} {\bibfield {journal} {\bibinfo {journal} {New + Journal of Physics}\ }\textbf {\bibinfo {volume} {19}},\ \bibinfo {pages} + {113017} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Nielsen}\ and\ \citenamefont + {Chuang}(2010)}]{nielsen_chuang_2010}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Michael~A.}\ + \bibnamefont {Nielsen}}\ and\ \bibinfo {author} {\bibfnamefont {Isaac~L.}\ + \bibnamefont {Chuang}},\ }\href {\doibase 10.1017/CBO9780511976667} {\emph + {\bibinfo {title} {Quantum Computation and Quantum Information: 10th + Anniversary Edition}}}\ (\bibinfo {publisher} {Cambridge University Press},\ + \bibinfo {year} {2010})\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Häffner}\ \emph {et~al.}(2005)\citenamefont + {Häffner}, \citenamefont {Hänsel}, \citenamefont {Roos}, \citenamefont + {Benhelm}, \citenamefont {Chek-al kar}, \citenamefont {Chwalla}, + \citenamefont {Körber}, \citenamefont {Rapol}, \citenamefont {Riebe}, + \citenamefont {Schmidt}, \citenamefont {Becher}, \citenamefont {Gühne}, + \citenamefont {Dür},\ and\ \citenamefont {Blatt}}]{haffner_scalable_2005}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont + {Häffner}}, \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Hänsel}}, + \bibinfo {author} {\bibfnamefont {C.~F.}\ \bibnamefont {Roos}}, \bibinfo + {author} {\bibfnamefont {J.}~\bibnamefont {Benhelm}}, \bibinfo {author} + {\bibfnamefont {D.}~\bibnamefont {Chek-al kar}}, \bibinfo {author} + {\bibfnamefont {M.}~\bibnamefont {Chwalla}}, \bibinfo {author} {\bibfnamefont + {T.}~\bibnamefont {Körber}}, \bibinfo {author} {\bibfnamefont {U.~D.}\ + \bibnamefont {Rapol}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont + {Riebe}}, \bibinfo {author} {\bibfnamefont {P.~O.}\ \bibnamefont {Schmidt}}, + \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Becher}}, \bibinfo + {author} {\bibfnamefont {O.}~\bibnamefont {Gühne}}, \bibinfo {author} + {\bibfnamefont {W.}~\bibnamefont {Dür}}, \ and\ \bibinfo {author} + {\bibfnamefont {R.}~\bibnamefont {Blatt}},\ }\bibfield {title} {\enquote + {\bibinfo {title} {Scalable multiparticle entanglement of trapped ions},}\ + }\href {\doibase 10.1038/nature04279} {\bibfield {journal} {\bibinfo + {journal} {Nature}\ }\textbf {\bibinfo {volume} {438}},\ \bibinfo {pages} + {643--646} (\bibinfo {year} {2005})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Gross}\ \emph {et~al.}(2010)\citenamefont {Gross}, + \citenamefont {Liu}, \citenamefont {Flammia}, \citenamefont {Becker},\ and\ + \citenamefont {Eisert}}]{PhysRevLett.105.150401}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {David}\ \bibnamefont + {Gross}}, \bibinfo {author} {\bibfnamefont {Yi-Kai}\ \bibnamefont {Liu}}, + \bibinfo {author} {\bibfnamefont {Steven~T.}\ \bibnamefont {Flammia}}, + \bibinfo {author} {\bibfnamefont {Stephen}\ \bibnamefont {Becker}}, \ and\ + \bibinfo {author} {\bibfnamefont {Jens}\ \bibnamefont {Eisert}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Quantum state tomography via compressed + sensing},}\ }\href {\doibase 10.1103/PhysRevLett.105.150401} {\bibfield + {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo + {volume} {105}},\ \bibinfo {pages} {150401} (\bibinfo {year} + {2010})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Flammia}\ \emph {et~al.}(2012)\citenamefont + {Flammia}, \citenamefont {Gross}, \citenamefont {Liu},\ and\ \citenamefont + {Eisert}}]{Flammia_2012}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Steven~T}\ + \bibnamefont {Flammia}}, \bibinfo {author} {\bibfnamefont {David}\ + \bibnamefont {Gross}}, \bibinfo {author} {\bibfnamefont {Yi-Kai}\ + \bibnamefont {Liu}}, \ and\ \bibinfo {author} {\bibfnamefont {Jens}\ + \bibnamefont {Eisert}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Quantum tomography via compressed sensing: error bounds, sample complexity + and efficient estimators},}\ }\href {\doibase 10.1088/1367-2630/14/9/095022} + {\bibfield {journal} {\bibinfo {journal} {New Journal of Physics}\ }\textbf + {\bibinfo {volume} {14}},\ \bibinfo {pages} {095022} (\bibinfo {year} + {2012})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Shabani}\ \emph {et~al.}(2011)\citenamefont + {Shabani}, \citenamefont {Kosut}, \citenamefont {Mohseni}, \citenamefont + {Rabitz}, \citenamefont {Broome}, \citenamefont {Almeida}, \citenamefont + {Fedrizzi},\ and\ \citenamefont {White}}]{PhysRevLett.106.100401}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont + {Shabani}}, \bibinfo {author} {\bibfnamefont {R.~L.}\ \bibnamefont {Kosut}}, + \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Mohseni}}, \bibinfo + {author} {\bibfnamefont {H.}~\bibnamefont {Rabitz}}, \bibinfo {author} + {\bibfnamefont {M.~A.}\ \bibnamefont {Broome}}, \bibinfo {author} + {\bibfnamefont {M.~P.}\ \bibnamefont {Almeida}}, \bibinfo {author} + {\bibfnamefont {A.}~\bibnamefont {Fedrizzi}}, \ and\ \bibinfo {author} + {\bibfnamefont {A.~G.}\ \bibnamefont {White}},\ }\bibfield {title} {\enquote + {\bibinfo {title} {Efficient measurement of quantum dynamics via compressive + sensing},}\ }\href {\doibase 10.1103/PhysRevLett.106.100401} {\bibfield + {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo + {volume} {106}},\ \bibinfo {pages} {100401} (\bibinfo {year} + {2011})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Riofrío}\ \emph {et~al.}(2017)\citenamefont + {Riofrío}, \citenamefont {Gross}, \citenamefont {Flammia}, \citenamefont + {Monz}, \citenamefont {Nigg}, \citenamefont {Blatt},\ and\ \citenamefont + {Eisert}}]{riofrio_experimental_2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {C.~A.}\ \bibnamefont + {Riofrío}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Gross}}, + \bibinfo {author} {\bibfnamefont {S.~T.}\ \bibnamefont {Flammia}}, \bibinfo + {author} {\bibfnamefont {T.}~\bibnamefont {Monz}}, \bibinfo {author} + {\bibfnamefont {D.}~\bibnamefont {Nigg}}, \bibinfo {author} {\bibfnamefont + {R.}~\bibnamefont {Blatt}}, \ and\ \bibinfo {author} {\bibfnamefont + {J.}~\bibnamefont {Eisert}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Experimental quantum compressed sensing for a seven-qubit system},}\ }\href + {\doibase 10.1038/ncomms15305} {\bibfield {journal} {\bibinfo {journal} + {Nature Communications}\ }\textbf {\bibinfo {volume} {8}},\ \bibinfo {pages} + {15305} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {T\'oth}\ \emph {et~al.}(2010)\citenamefont {T\'oth}, + \citenamefont {Wieczorek}, \citenamefont {Gross}, \citenamefont {Krischek}, + \citenamefont {Schwemmer},\ and\ \citenamefont + {Weinfurter}}]{PhysRevLett.105.250403}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont + {T\'oth}}, \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Wieczorek}}, + \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Gross}}, \bibinfo + {author} {\bibfnamefont {R.}~\bibnamefont {Krischek}}, \bibinfo {author} + {\bibfnamefont {C.}~\bibnamefont {Schwemmer}}, \ and\ \bibinfo {author} + {\bibfnamefont {H.}~\bibnamefont {Weinfurter}},\ }\bibfield {title} + {\enquote {\bibinfo {title} {Permutationally invariant quantum tomography},}\ + }\href {\doibase 10.1103/PhysRevLett.105.250403} {\bibfield {journal} + {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {105}},\ + \bibinfo {pages} {250403} (\bibinfo {year} {2010})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Moroder}\ \emph {et~al.}(2012)\citenamefont + {Moroder}, \citenamefont {Hyllus}, \citenamefont {T{\'{o}}th}, \citenamefont + {Schwemmer}, \citenamefont {Niggebaum}, \citenamefont {Gaile}, \citenamefont + {Gühne},\ and\ \citenamefont {Weinfurter}}]{Moroder_2012}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Tobias}\ \bibnamefont + {Moroder}}, \bibinfo {author} {\bibfnamefont {Philipp}\ \bibnamefont + {Hyllus}}, \bibinfo {author} {\bibfnamefont {G{\'{e}}za}\ \bibnamefont + {T{\'{o}}th}}, \bibinfo {author} {\bibfnamefont {Christian}\ \bibnamefont + {Schwemmer}}, \bibinfo {author} {\bibfnamefont {Alexander}\ \bibnamefont + {Niggebaum}}, \bibinfo {author} {\bibfnamefont {Stefanie}\ \bibnamefont + {Gaile}}, \bibinfo {author} {\bibfnamefont {Otfried}\ \bibnamefont {Gühne}}, + \ and\ \bibinfo {author} {\bibfnamefont {Harald}\ \bibnamefont + {Weinfurter}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Permutationally invariant state reconstruction},}\ }\href {\doibase + 10.1088/1367-2630/14/10/105001} {\bibfield {journal} {\bibinfo {journal} + {New Journal of Physics}\ }\textbf {\bibinfo {volume} {14}},\ \bibinfo + {pages} {105001} (\bibinfo {year} {2012})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Cramer}\ \emph {et~al.}(2009)\citenamefont {Cramer}, + \citenamefont {Plenio}, \citenamefont {Flammia}, \citenamefont {Somma}, + \citenamefont {Gross}, \citenamefont {Bartlett}, \citenamefont + {Landon-Cardinal}, \citenamefont {Poulin},\ and\ \citenamefont + {Liu}}]{cramer2009efficient}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {M}~\bibnamefont + {Cramer}}, \bibinfo {author} {\bibfnamefont {MB}~\bibnamefont {Plenio}}, + \bibinfo {author} {\bibfnamefont {ST}~\bibnamefont {Flammia}}, \bibinfo + {author} {\bibfnamefont {R}~\bibnamefont {Somma}}, \bibinfo {author} + {\bibfnamefont {D}~\bibnamefont {Gross}}, \bibinfo {author} {\bibfnamefont + {SD}~\bibnamefont {Bartlett}}, \bibinfo {author} {\bibfnamefont + {O}~\bibnamefont {Landon-Cardinal}}, \bibinfo {author} {\bibfnamefont + {D}~\bibnamefont {Poulin}}, \ and\ \bibinfo {author} {\bibfnamefont + {YK}~\bibnamefont {Liu}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Efficient quantum state tomography.}}\ }\href + {http://www.nature.com/articles/ncomms1147} {\bibfield {journal} {\bibinfo + {journal} {Nature communications}\ }\textbf {\bibinfo {volume} {1}},\ + \bibinfo {pages} {149} (\bibinfo {year} {2009})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Baumgratz}\ \emph {et~al.}(2013)\citenamefont + {Baumgratz}, \citenamefont {Gross}, \citenamefont {Cramer},\ and\ + \citenamefont {Plenio}}]{MPOtomo}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont + {Baumgratz}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Gross}}, + \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Cramer}}, \ and\ \bibinfo + {author} {\bibfnamefont {M.~B.}\ \bibnamefont {Plenio}},\ }\bibfield {title} + {\enquote {\bibinfo {title} {Scalable reconstruction of density matrices},}\ + }\href {\doibase 10.1103/PhysRevLett.111.020401} {\bibfield {journal} + {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {111}},\ + \bibinfo {pages} {020401} (\bibinfo {year} {2013})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Lanyon}\ \emph {et~al.}(2017)\citenamefont {Lanyon}, + \citenamefont {Maier}, \citenamefont {Holz{\"{a}}pfel}, \citenamefont + {Baumgratz}, \citenamefont {Hempel}, \citenamefont {Jurcevic}, \citenamefont + {Dhand}, \citenamefont {Buyskikh}, \citenamefont {Daley}, \citenamefont + {Cramer}, \citenamefont {Plenio}, \citenamefont {Blatt},\ and\ \citenamefont + {Roos}}]{Lanyon2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {B.~P.}\ \bibnamefont + {Lanyon}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Maier}}, + \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Holz{\"{a}}pfel}}, + \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Baumgratz}}, \bibinfo + {author} {\bibfnamefont {C.}~\bibnamefont {Hempel}}, \bibinfo {author} + {\bibfnamefont {P.}~\bibnamefont {Jurcevic}}, \bibinfo {author} + {\bibfnamefont {I.}~\bibnamefont {Dhand}}, \bibinfo {author} {\bibfnamefont + {A.~S.}\ \bibnamefont {Buyskikh}}, \bibinfo {author} {\bibfnamefont {A.~J.}\ + \bibnamefont {Daley}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont + {Cramer}}, \bibinfo {author} {\bibfnamefont {M.~B.}\ \bibnamefont {Plenio}}, + \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Blatt}}, \ and\ \bibinfo + {author} {\bibfnamefont {C.~F.}\ \bibnamefont {Roos}},\ }\bibfield {title} + {\enquote {\bibinfo {title} {{Efficient tomography of a quantum many-body + system}},}\ }\href {\doibase 10.1038/nphys4244} {\bibfield {journal} + {\bibinfo {journal} {Nature Physics}\ }\textbf {\bibinfo {volume} {13}},\ + \bibinfo {pages} {1158--1162} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Wang}\ \emph {et~al.}(2020)\citenamefont {Wang}, + \citenamefont {Han}, \citenamefont {Wang}, \citenamefont {Li}, \citenamefont + {Mu}, \citenamefont {Fan},\ and\ \citenamefont {Wang}}]{LeiWang2020}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Jun}\ \bibnamefont + {Wang}}, \bibinfo {author} {\bibfnamefont {Zhao-Yu}\ \bibnamefont {Han}}, + \bibinfo {author} {\bibfnamefont {Song-Bo}\ \bibnamefont {Wang}}, \bibinfo + {author} {\bibfnamefont {Zeyang}\ \bibnamefont {Li}}, \bibinfo {author} + {\bibfnamefont {Liang-Zhu}\ \bibnamefont {Mu}}, \bibinfo {author} + {\bibfnamefont {Heng}\ \bibnamefont {Fan}}, \ and\ \bibinfo {author} + {\bibfnamefont {Lei}\ \bibnamefont {Wang}},\ }\bibfield {title} {\enquote + {\bibinfo {title} {Scalable quantum tomography with fidelity estimation},}\ + }\href {\doibase 10.1103/PhysRevA.101.032321} {\bibfield {journal} {\bibinfo + {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {101}},\ \bibinfo + {pages} {032321} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Ackley}\ \emph {et~al.}(1985)\citenamefont {Ackley}, + \citenamefont {Hinton},\ and\ \citenamefont {Sejnowski}}]{Ackley85}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {David~H.}\ + \bibnamefont {Ackley}}, \bibinfo {author} {\bibfnamefont {Geoffrey~E.}\ + \bibnamefont {Hinton}}, \ and\ \bibinfo {author} {\bibfnamefont + {Terrence~J.}\ \bibnamefont {Sejnowski}},\ }\bibfield {title} {\enquote + {\bibinfo {title} {A learning algorithm for boltzmann machines},}\ }\href + {\doibase https://doi.org/10.1016/S0364-0213(85)80012-4} {\bibfield + {journal} {\bibinfo {journal} {Cognitive Science}\ }\textbf {\bibinfo + {volume} {9}},\ \bibinfo {pages} {147 -- 169} (\bibinfo {year} + {1985})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Smolensky}(1986)}]{Smolensky1986}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont + {Smolensky}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Parallel + distributed processing: Explorations in the microstructure of cognition, vol. + 1},}\ }in\ \href {http://dl.acm.org/citation.cfm?id=104279.104290} {\emph + {\bibinfo {booktitle} {Parallel Distributed Processing}}},\ \bibinfo {editor} + {edited by\ \bibinfo {editor} {\bibfnamefont {David~E.}\ \bibnamefont + {Rumelhart}}, \bibinfo {editor} {\bibfnamefont {James~L.}\ \bibnamefont + {McClelland}}, \ and\ \bibinfo {editor} {\bibfnamefont {CORPORATE}\ + \bibnamefont {PDP Research~Group}}}\ (\bibinfo {publisher} {MIT Press},\ + \bibinfo {address} {Cambridge, MA, USA},\ \bibinfo {year} {1986})\ Chap.\ + \bibinfo {chapter} {Information Processing in Dynamical Systems: Foundations + of Harmony Theory}, pp.\ \bibinfo {pages} {194--281}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Little}(1974)}]{Little74}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {W.A.}\ \bibnamefont + {Little}},\ }\bibfield {title} {\enquote {\bibinfo {title} {The existence of + persistent states in the brain},}\ }\href {\doibase + https://doi.org/10.1016/0025-5564(74)90031-5} {\bibfield {journal} {\bibinfo + {journal} {Mathematical Biosciences}\ }\textbf {\bibinfo {volume} {19}},\ + \bibinfo {pages} {101 -- 120} (\bibinfo {year} {1974})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Little}\ and\ \citenamefont {Shaw}(1978)}]{Little78}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {W.A.}\ \bibnamefont + {Little}}\ and\ \bibinfo {author} {\bibfnamefont {Gordon~L.}\ \bibnamefont + {Shaw}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Analytic study of + the memory storage capacity of a neural network},}\ }\href {\doibase + https://doi.org/10.1016/0025-5564(78)90058-5} {\bibfield {journal} {\bibinfo + {journal} {Mathematical Biosciences}\ }\textbf {\bibinfo {volume} {39}},\ + \bibinfo {pages} {281 -- 290} (\bibinfo {year} {1978})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Hopfield}(1982)}]{Hopfield82}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {J~J}\ \bibnamefont + {Hopfield}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Neural + networks and physical systems with emergent collective computational + abilities},}\ }\href {\doibase 10.1073/pnas.79.8.2554} {\bibfield {journal} + {\bibinfo {journal} {Proceedings of the National Academy of Sciences}\ + }\textbf {\bibinfo {volume} {79}},\ \bibinfo {pages} {2554--2558} (\bibinfo + {year} {1982})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Le~Roux}\ and\ \citenamefont + {Bengio}(2008)}]{LeRoux2008}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Nicolas}\ \bibnamefont + {Le~Roux}}\ and\ \bibinfo {author} {\bibfnamefont {Yoshua}\ \bibnamefont + {Bengio}},\ }\bibfield {title} {\enquote {\bibinfo {title} {Representational + power of restricted boltzmann machines and deep belief networks},}\ }\href + {\doibase 10.1162/neco.2008.04-07-510} {\bibfield {journal} {\bibinfo + {journal} {Neural Computation}\ }\textbf {\bibinfo {volume} {20}},\ \bibinfo + {pages} {1631--1649} (\bibinfo {year} {2008})},\ \Eprint + {http://arxiv.org/abs/https://doi.org/10.1162/neco.2008.04-07-510} + {https://doi.org/10.1162/neco.2008.04-07-510} \BibitemShut {NoStop}% +\bibitem [{\citenamefont {Kullback}\ and\ \citenamefont + {Leibler}(1951)}]{Kullback:1951aa}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont + {Kullback}}\ and\ \bibinfo {author} {\bibfnamefont {R.~A.}\ \bibnamefont + {Leibler}},\ }\bibfield {title} {\enquote {\bibinfo {title} {On information + and sufficiency},}\ }\href {\doibase 10.1214/aoms/1177729694} {\bibfield + {journal} {\bibinfo {journal} {Ann. Math. Statist.}\ }\textbf {\bibinfo + {volume} {22}},\ \bibinfo {pages} {79--86} (\bibinfo {year} + {1951})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Bravyi}}\ \emph {et~al.}(2006)\citenamefont + {{Bravyi}}, \citenamefont {{DiVincenzo}}, \citenamefont {{Oliveira}},\ and\ + \citenamefont {{Terhal}}}]{Bravyi2006}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Sergey}\ \bibnamefont + {{Bravyi}}}, \bibinfo {author} {\bibfnamefont {David~P.}\ \bibnamefont + {{DiVincenzo}}}, \bibinfo {author} {\bibfnamefont {Roberto~I.}\ \bibnamefont + {{Oliveira}}}, \ and\ \bibinfo {author} {\bibfnamefont {Barbara~M.}\ + \bibnamefont {{Terhal}}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {{The Complexity of Stoquastic Local Hamiltonian Problems}},}\ }\href@noop {} + {\bibfield {journal} {\bibinfo {journal} {arXiv e-prints}\ ,\ \bibinfo + {eid} {quant-ph/0606140}} (\bibinfo {year} {2006})},\ \Eprint + {http://arxiv.org/abs/quant-ph/0606140} {arXiv:quant-ph/0606140 [quant-ph]} + \BibitemShut {NoStop}% +\bibitem [{\citenamefont {{Carleo}}\ \emph {et~al.}(2019)\citenamefont + {{Carleo}}, \citenamefont {{Choo}}, \citenamefont {{Hofmann}}, \citenamefont + {{Smith}}, \citenamefont {{Westerhout}}, \citenamefont {{Alet}}, + \citenamefont {{Davis}}, \citenamefont {{Efthymiou}}, \citenamefont + {{Glasser}}, \citenamefont {{Lin}}, \citenamefont {{Mauri}}, \citenamefont + {{Mazzola}}, \citenamefont {{Mendl}}, \citenamefont {{van Nieuwenburg}}, + \citenamefont {{O'Reilly}}, \citenamefont {{Th{\'e}veniaut}}, \citenamefont + {{Torlai}},\ and\ \citenamefont {{Wietek}}}]{netket}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Giuseppe}\ + \bibnamefont {{Carleo}}}, \bibinfo {author} {\bibfnamefont {Kenny}\ + \bibnamefont {{Choo}}}, \bibinfo {author} {\bibfnamefont {Damian}\ + \bibnamefont {{Hofmann}}}, \bibinfo {author} {\bibfnamefont {James E.~T.}\ + \bibnamefont {{Smith}}}, \bibinfo {author} {\bibfnamefont {Tom}\ \bibnamefont + {{Westerhout}}}, \bibinfo {author} {\bibfnamefont {Fabien}\ \bibnamefont + {{Alet}}}, \bibinfo {author} {\bibfnamefont {Emily~J.}\ \bibnamefont + {{Davis}}}, \bibinfo {author} {\bibfnamefont {Stavros}\ \bibnamefont + {{Efthymiou}}}, \bibinfo {author} {\bibfnamefont {Ivan}\ \bibnamefont + {{Glasser}}}, \bibinfo {author} {\bibfnamefont {Sheng-Hsuan}\ \bibnamefont + {{Lin}}}, \bibinfo {author} {\bibfnamefont {Marta}\ \bibnamefont {{Mauri}}}, + \bibinfo {author} {\bibfnamefont {Guglielmo}\ \bibnamefont {{Mazzola}}}, + \bibinfo {author} {\bibfnamefont {Christian~B.}\ \bibnamefont {{Mendl}}}, + \bibinfo {author} {\bibfnamefont {Evert}\ \bibnamefont {{van Nieuwenburg}}}, + \bibinfo {author} {\bibfnamefont {Ossian}\ \bibnamefont {{O'Reilly}}}, + \bibinfo {author} {\bibfnamefont {Hugo}\ \bibnamefont {{Th{\'e}veniaut}}}, + \bibinfo {author} {\bibfnamefont {Giacomo}\ \bibnamefont {{Torlai}}}, \ and\ + \bibinfo {author} {\bibfnamefont {Alexander}\ \bibnamefont {{Wietek}}},\ + }\bibfield {title} {\enquote {\bibinfo {title} {{NetKet: A Machine Learning + Toolkit for Many-Body Quantum Systems}},}\ }\href@noop {} {\bibfield + {journal} {\bibinfo {journal} {arXiv e-prints}\ ,\ \bibinfo {eid} + {arXiv:1904.00031}} (\bibinfo {year} {2019})},\ \Eprint + {http://arxiv.org/abs/1904.00031} {arXiv:1904.00031 [quant-ph]} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {{Zeiler}}(2012)}]{2012arXiv1212.5701Z}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Matthew~D.}\ + \bibnamefont {{Zeiler}}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {{ADADELTA: An Adaptive Learning Rate Method}},}\ }\href@noop {} {\bibfield + {journal} {\bibinfo {journal} {arXiv e-prints}\ ,\ \bibinfo {eid} + {arXiv:1212.5701}} (\bibinfo {year} {2012})},\ \Eprint + {http://arxiv.org/abs/1212.5701} {arXiv:1212.5701 [cs.LG]} \BibitemShut + {NoStop}% +\bibitem [{\citenamefont {Kandala}\ \emph {et~al.}(2017)\citenamefont + {Kandala}, \citenamefont {Mezzacapo}, \citenamefont {Temme}, \citenamefont + {Takita}, \citenamefont {Brink}, \citenamefont {Chow},\ and\ \citenamefont + {Gambetta}}]{kandala_hardware-efficient_2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Abhinav}\ \bibnamefont + {Kandala}}, \bibinfo {author} {\bibfnamefont {Antonio}\ \bibnamefont + {Mezzacapo}}, \bibinfo {author} {\bibfnamefont {Kristan}\ \bibnamefont + {Temme}}, \bibinfo {author} {\bibfnamefont {Maika}\ \bibnamefont {Takita}}, + \bibinfo {author} {\bibfnamefont {Markus}\ \bibnamefont {Brink}}, \bibinfo + {author} {\bibfnamefont {Jerry~M.}\ \bibnamefont {Chow}}, \ and\ \bibinfo + {author} {\bibfnamefont {Jay~M.}\ \bibnamefont {Gambetta}},\ }\bibfield + {title} {\enquote {\bibinfo {title} {Hardware-efficient variational quantum + eigensolver for small molecules and quantum magnets},}\ }\href {\doibase + 10.1038/nature23879} {\bibfield {journal} {\bibinfo {journal} {Nature}\ + }\textbf {\bibinfo {volume} {549}},\ \bibinfo {pages} {242--246} (\bibinfo + {year} {2017})}\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Becca}\ and\ \citenamefont + {Sorella}(2017)}]{becca_sorella_2017}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Federico}\ + \bibnamefont {Becca}}\ and\ \bibinfo {author} {\bibfnamefont {Sandro}\ + \bibnamefont {Sorella}},\ }\href {\doibase 10.1017/9781316417041} {\emph + {\bibinfo {title} {Quantum Monte Carlo Approaches for Correlated Systems}}}\ + (\bibinfo {publisher} {Cambridge University Press},\ \bibinfo {year} + {2017})\BibitemShut {NoStop}% +\bibitem [{\citenamefont {Cho}\ \emph {et~al.}(2014)\citenamefont {Cho}, + \citenamefont {van Merri{\"e}nboer}, \citenamefont {Gulcehre}, \citenamefont + {Bahdanau}, \citenamefont {Bougares}, \citenamefont {Schwenk},\ and\ + \citenamefont {Bengio}}]{cho-etal-2014-learning}% + \BibitemOpen + \bibfield {author} {\bibinfo {author} {\bibfnamefont {Kyunghyun}\ + \bibnamefont {Cho}}, \bibinfo {author} {\bibfnamefont {Bart}\ \bibnamefont + {van Merri{\"e}nboer}}, \bibinfo {author} {\bibfnamefont {Caglar}\ + \bibnamefont {Gulcehre}}, \bibinfo {author} {\bibfnamefont {Dzmitry}\ + \bibnamefont {Bahdanau}}, \bibinfo {author} {\bibfnamefont {Fethi}\ + \bibnamefont {Bougares}}, \bibinfo {author} {\bibfnamefont {Holger}\ + \bibnamefont {Schwenk}}, \ and\ \bibinfo {author} {\bibfnamefont {Yoshua}\ + \bibnamefont {Bengio}},\ }\bibfield {title} {\enquote {\bibinfo {title} + {Learning phrase representations using {RNN} encoder{--}decoder for + statistical machine translation},}\ }in\ \href {\doibase 10.3115/v1/D14-1179} + {\emph {\bibinfo {booktitle} {Proceedings of the 2014 Conference on Empirical + Methods in Natural Language Processing ({EMNLP})}}}\ (\bibinfo {publisher} + {Association for Computational Linguistics},\ \bibinfo {address} {Doha, + Qatar},\ \bibinfo {year} {2014})\ pp.\ \bibinfo {pages} + {1724--1734}\BibitemShut {NoStop}% +\end{thebibliography}% diff --git a/doc/Articles/main.tex b/doc/Articles/main.tex new file mode 100644 index 000000000..4bde84092 --- /dev/null +++ b/doc/Articles/main.tex @@ -0,0 +1,734 @@ +\documentclass[twocolumn,english,reprint,superscriptaddress,longbibliography,pra]{revtex4-1} +\renewcommand{\familydefault}{\rmdefault} +\usepackage[T1]{fontenc} +\usepackage[latin9]{inputenc} +\setcounter{secnumdepth}{3} +\usepackage{color} +\usepackage{bm} +\usepackage{amstext} +\usepackage{amssymb} +\usepackage{graphicx} +\usepackage{booktabs} +\usepackage{tabularx} +\usepackage{amsmath,mathrsfs} +\usepackage{enumitem} +\makeatletter +\usepackage{multibib} +\usepackage{listings} +\usepackage{mathtools} +\usepackage{braket} + + +\DeclareMathOperator*{\argmin}{argmin} + +\usepackage{babel} +\usepackage{bbold} +\usepackage{mathrsfs} +\usepackage{hyperref} +\hypersetup{ + linktocpage = true, + colorlinks, + citecolor=blue, + filecolor=black, + linkcolor=blue, + urlcolor=blue +} +\newcommand{\BeH}{BeH$_2$} +\newcommand{\tr}{\textrm{Tr}} +\newcommand{\FIG}{\textcolor{blue}{FIGURE\:}} +\newcommand{\REF}{[\textcolor{red}{ref}]} + + + +\usepackage{xcolor} +\definecolor{codegreen}{rgb}{0,0.6,0} +\definecolor{codegray}{rgb}{0.5,0.5,0.5} +\definecolor{codepurple}{rgb}{0.58,0,0.82} +\definecolor{backcolour}{rgb}{0.95,0.95,0.92} +\lstdefinestyle{mystyle}{ + backgroundcolor=\color{backcolour}, + commentstyle=\color{codegreen}, + keywordstyle=\color{magenta}, + numberstyle=\tiny\color{codegray}, + stringstyle=\color{codepurple}, + basicstyle=\ttfamily\footnotesize, + breakatwhitespace=false, + breaklines=true, + captionpos=b, + keepspaces=true, + numbers=left, + numbersep=5pt, + showspaces=false, + showstringspaces=false, + showtabs=false, + tabsize=2 +} +\lstset{style=mystyle} +\usepackage{xpatch} + +\makeatletter +\patchcmd{\@ssect@ltx} + {\addcontentsline{toc}{#1}{\protect\numberline{}#8}} + {} + {} + {} +\makeatother + + +\begin{document} + +\title{Neural networks in quantum many-body physics: a hands-on tutorial} + +\author{Juan Carrasquilla} +\affiliation{Vector Institute, MaRS Centre, Toronto, Ontario, M5G 1M1, Canada} +\author{Giacomo Torlai} +\email{gttorlai@amazon.com} +\thanks{\\This work has been done before Giacomo Torlai joined Amazon.} +\affiliation{AWS Center for Quantum Computing, Pasadena, CA 91125, USA} +\affiliation{Center for Computational Quantum Physics, Flatiron Institute, New York, NY 10010, USA} + + + +\begin{abstract} +Over the past years, machine learning has emerged as a powerful computational tool to tackle complex problems over a broad range of scientific disciplines. In particular, artificial neural networks have been successfully deployed to mitigate the exponential complexity often encountered in quantum many-body physics, the study of properties of quantum systems built out of a large number of interacting particles. In this Article, we overview some applications of machine learning in condensed matter physics and quantum information, with particular emphasis on hands-on tutorials serving as a quick-start for a newcomer to the field. We present supervised machine learning with convolutional neural networks to learn a phase transition, unsupervised learning with restricted Boltzmann machines to perform quantum tomography, and variational Monte Carlo with recurrent neural-networks for approximating the ground state of a many-body Hamiltonian. We briefly review the key ingredients of each algorithm and their corresponding neural-network implementation, and show numerical experiments for a system of interacting Rydberg atoms in two dimensions. +\end{abstract} + +\maketitle + +\section{Introduction} + +Quantum many-body physics refers to the mathematical framework to study the collective behavior of large numbers of interacting particles. The emerging cooperative phenomena that result from seemingly simple interactions can produce an astounding variety of phases of matter such as conventional metals and magnetically ordered states, as well as unanticipated states including high-temperature superconductivity, strange metals, and spin liquids~\cite{Xiao:803748}. In addition to naturally occurring quantum systems, many-body physics studies synthetic quantum matter, e.g., ultracold atoms, superconducting qubits, and trapped ions, which simultaneously reveal new phenomena in highly controlled laboratory settings and advances the development of quantum computers and other quantum information processing devices. + +In spite of the simplicity of the physical laws that govern such multi-particle quantum objects, the theoretical and experimental analysis of these systems confront us with complexities which are ultimately rooted in the "curse of dimensionality" associated with the exponential explosion of the size of the space where quantum many-body states live in. Traditionally, the study of many-body systems is performed with the help of tools designed to circumvent this dimensionality explosion and produce a succinct, low-dimensional description that captures the essential aspects of a quantum system. Such descriptions arise from the analysis of data generated in a wide range of theoretical, computational, and experimental devices. These include numerical simulations of model Hamiltonians based on quantum Monte Carlo or variational algorithms, but also experimental arrays of complex electronic-structure images obtained from spectroscopic imaging scanning tunnelling microscopy, or measurements of quantum states prepared on a physical quantum computing platform. + +Machine learning, already explored as a tool in several research areas in physics~\cite{RevModPhys.91.045002}, offers a set of alternative approaches to the study of quantum many-body systems in experiments and numerical simulations~\cite{doi:10.1080/23746149.2020.1797528,annurev-conmatphys-031119-050651}. The resurgence of activity at the intersection between physics and machine learning is in part due to a series of scientific breakthroughs in computer vision and natural language processing. Such progress has led to a burst of research where neural networks have been repurposed to tackle fundamental questions in condensed matter physics, quantum computing, statistical physics, and atomic, molecular and optical physics. Machine learning, and in particular deep neural networks, have been used to identify phases of matter in numerical simulations and experiments~\cite{carrasquilla2017nature, evert2017nature, torlai_learning_2016, leiwang2016, chng2017, broecker2017, eun-ah2017, dassarma2017, neupeurt2017, yi-ting2018, huembeli2018,PhysRevB.99.060404,PhysRevB.99.104410,Zhang_MLcuprates,Bohrdt2018,Rem2018,PhysRevLett.122.210503}, +to increase the performance of Monte Carlo simulations~\cite{huang2017,junwei2017, xiao_yan2017, inack2018, parolini2019, pilati2019, mcnaughton2020, albergo2019}, to accurately describe the state of classical~\cite{Wu_2019} and quantum systems~\cite{androsiuk1993,LAGARIS19971, Carleo_2017, zi2018, Di_Luo, pfau2019abinitio, hermann2019deep, PhysRevLett.122.250502, PhysRevLett.122.250501,PhysRevLett.122.250503,PhysRevB.99.214306, choo_fermionicnqs2020, PhysRevLett.124.020503, RNNWF_2020, roth2020iterative}, to develop novel quantum control strategies~\cite{PhysRevX.8.031086,PhysRevX.8.031084,PhysRevLett.122.020601,niu_universal_2019,2020arXiv201003655Y,coopmans2020}, to perform quantum tomography~\cite{torlai_Tomo,rocchetto,Torlai_latent,2018arXiv181206693Q,carrasquilla_povm,biamonte_qst,torlai_rydberg19,xin_local-measurement-based_2019,Sehayek2019,torlai_chemistry,PhysRevA.102.022412,NoriGAN,Tiunov:20,Cha2020,PhysRevA.102.042604,DeVlugt2020,2020arXiv200907601S,torlai_QPT,morawetz2020,Nori2020}, to accelerate density functional theory calculations~\cite{PhysRevLett.108.253002,doi:10.1063/1.4834075,PhysRevB.94.245129,brockherde_bypassing_2017,PhysRevA.100.022512,PhysRevLett.125.076402,PhysRevResearch.2.033388}, to develop and elucidate renormalization group analyses~\cite{2014arXiv1410.3831M,koch-janusz_mutual_2018,PhysRevE.97.053304,PhysRevLett.121.260601,PhysRevResearch.2.023369,2020arXiv201005703C}, to devise quantum error correction protocols~\cite{torlai_neural_2016,krastanov_deep_2017,Varsamopoulos_2017,Baireuther2018machinelearning,Chamberland_2018,Breuckmann2018scalableneural,Nautrup2019optimizingquantum,PhysRevLett.122.200501,PhysRevA.99.052351,Andreasson2019quantumerror,Evert2020QEC,Ni2020neuralnetwork,PhysRevResearch.2.033399,PhysRevResearch.2.023230}, among many other examples~\cite{PhysRevB.97.045153,Seif_2018,Melnikov1221,Dunjko_2018,PhysRevE.99.062106, 2019arXiv191211052C,PhysRevB.99.075113,PhysRevLett.124.010508, PhysRevX.10.011006,2020arXiv200600712H,2020arXiv200905580L,2020arXiv201014510L}. + +Such an explosion of activity indicates that machine learning techniques may soon become commonplace in quantum many-body physics research, both in experiments and numerical simulation. These clear trends call for the development of resources to stimulate researchers to familiarize with the wealth of concepts, intuition, algorithms, hardware, software, and research culture entailed by the adoption of machine learning and neural networks in physics research. Here, we take a step forward in this direction and develop a set of hands-on tutorials focused on a set of recent prototypical examples of applications of neural network technology to problems in statistical physics, condensed matter and quantum computing. + +\subsection*{Outline} +The Article is organized as follows. Starting with a preliminary discussion, we introduce in Sec~\ref{preliminariesA} some fundamental concepts in machine learning and neural networks. In Sec~\ref{preliminariesB} we present a concise description of the physical system studied in our numerical experiments, a two-dimensional array of interacting Rydberg atoms. In Sec~\ref{supervised} we discuss our first application, the classification of phases of matter with supervised machine learning of projective measurement data using a convolutional neural network, and demonstrate it on the quantum phase transition in the Rydberg atoms. In Sec~\ref{qst} we introduce quantum state tomography, and show how this problem can be phrased as an unsupervised machine learning task. Using the restricted Boltzmann machine, we show quantum tomography of the Rydberg ground states, as well as of the ground state of a small molecule from qubit measurement data. In Sec~\ref{vmc} we present the simulation of the ground state of a many-body Hamiltonian using variational Monte Carlo with a recurrent neural network wavefunction. For each of these applications, we also show the key components of the underlying software, with full code tutorials available in an external repository~\cite{coderepo}. + + +\section{Preliminaries} +\subsection{Machine learning with neural networks} +\label{preliminariesA} +Artificial intelligence, the scientific discipline that deals with the theory and development of computer programs with the ability to perform complex tasks, saw early success solving problems which are relatively straightforward to formalize in an abstract way. The solutions to this breed of problems are typically described by a list of very precise formal rules that computers can process efficiently. As remarkable example, computers have been beating humans at playing chess since 1997, due in part to the fact that chess involves a large set of formal rules. + +Modern machine learning, instead, deals with the challenge of automatizing the solution of real world tasks that may be easy for humans to process but that are hard to formally describe by simple rules. These techniques have spurred a recent revolution where algorithms trained using data have started to match humans' ability to recognize objects in an image, decipher speech or translate text to multiple languages, which are tasks that are difficult to formalize and articulate through simple rules. + +A key element behind these recent developments can be largely traced back to a series of breakthroughs in the development of powerful neural network models, where data is processed through the sequential combination of multiple nonlinear layers~\cite{Goodfellow-et-al-2016}. Such models solve a fundamental problem in learning real world tasks, namely the problem of automatically extracting knowledge from raw noisy data, rather than relying on hard-coded knowledge directly inscribed in the algorithms by a human. Neural networks automatize the construction of sets of increasingly complex representations of the data, which can be understood as the computational disentangling of complex concepts (e.g. an object in a cluttered image) out of simpler concepts (e.g. pixel values and basic shapes like edges). These representations, in turn, lead to solutions to learning tasks with unprecedented success. + +% ML tasks +\begin{figure*}[t] +\noindent \centering{}\includegraphics[width=2.05\columnwidth]{Fig1} +\caption{Rydberg atoms in a two-dimensional square array. ({\bf a}) Schematic representation of the phase diagram at a fixed value of the interaction $V=3$ MHz and $\Omega = 1$ MHz. At large and negative detuning, the system is in a disordered (paramagnetic) phase with all atoms in the ground state. At large and positive detuning, the atoms are found in a checkerboard pattern with N\'eel order. ({\bf b}) Snake-like geometry of the MPS path along the square lattice, used for the DMRG simulations. Ground state energy ({\bf c}) and the staggered magnetization (N\'eel order) ({\bf d}) as a function of the detuning $\delta$, for a $8\times8$ array ($V=3$ MHz). ({\bf e}) Absolute value of the average occupation number in momentum space $|n(\bm{k})|$ deep into the $Z_2$ ordered phase ($\delta=4$ MHz), showing a peak at $\bm{k}=(\pi,\pi)$, a signature of anti-ferromagnetic order. ({\bf f}) Energy gap $\Delta$ between the ground state and first excited state, detecting a quantum phase transition at detuning $\delta\approx1.3$.} +\label{Fig::1} +\end{figure*} + +For practical purposes, machine learning algorithms can be divided into the categories of supervised, unsupervised, and reinforcement learning, all of which have found applications to quantum many-body systems~\cite{doi:10.1080/23746149.2020.1797528}. While there is no formal difference between some of the algorithms in these categories when expressed in the language of probability~\cite{Goodfellow-et-al-2016,10.5555/1162264}, such a division is often used as a way to specify the details of the algorithms, the training setup, and the structure of the data sets involved. + +Supervised learning tasks aim at predicting a target output vector $\bm y$ associated with input vector $\bm x$, both of which can be discrete or continuous. The training data is thus a list of pairs of input/output tuples $\{ \bm x_i,\bm y_i\}_{i=1}^{M}$, where target output conveys that such a vector corresponds to the ideal output given the input vector~\cite{10.5555/1162264}. Starting with a training data set with $M$ entries, the learning algorithm outputs a function $\hat{\bm y} = f(\bm x)$ which estimates the output values for unseen input vectors $\bm x$. Examples of supervised learning include classification, where the objective is to assign each input vector to one of a set of discrete categories, and the task of +regression, where the output is a vector with continuous entries. Examples for classification and regression are respectively the problem of recognizing images of handwritten digits and the problem of determining the orbits of bodies around the sun from astronomical data. + +Unsupervised learning deals with the learning tasks where the training data is composed of a set of input vectors without a corresponding target output~\cite{10.5555/1162264}. These algorithms are typically used to discover hidden structure in the data sets. Examples of tasks in unsupervised learning problems include clustering, where the objective is to discover of groups of similar examples within the data, density estimation, where the objective is to estimate the underlying probability distribution associated with the data, as well as low-dimensional visualization of high-dimensional data algorithms, which depict complex data in two or three dimensions while trying to retain key spatial characteristics in the original data. + +Finally, reinforcement learning, although not discussed in this Article, develops algorithms dealing with the problem of discovering actions that maximize a numerical reward signal~\cite{10.5555/551283}. The learning algorithms are not necessarily directly exposed to examples of optimal actions. Instead, it must discover them by a process similar to a guided trial and error. Reinforcement learning augmented by deep neural networks has successfully learned policies from high-dimensional sensory input for game playing achieving human-level performance in several challenging games including Atari 2600~\cite{mnih_human-level_2015} as well as the board game Go~\cite{silver2016}. Likewise, reinforcement learning has been applied to the control of quantum systems~\cite{bukov2018,niu_universal_2019} as well as to the optimization of quantum error correction codes~\cite{Nautrup2019optimizingquantum,Andreasson2019quantumerror,Evert2020QEC}, one key ingredient in the development of fault-tolerant quantum computers. + + + +\subsection{Rydberg atoms in two dimensions} +\label{preliminariesB} +We demonstrate the machine learning algorithms discussed in this Article for a many-body system composed by interacting Rydberg atoms. Engineered arrays of cold Rydberg atoms are increasingly used for highly-controlled quantum simulations of strongly-interacting matter~\cite{Schauss1455,Endres2016,Labuhn,Bernien2017,keesling_quantum_2019,2020arXiv201212281E,2020arXiv201212268S}, as well as for quantum information processing~\cite{Levine2019}. We specifically consider a square array with linear dimension $L$ containing $N = L^2$ atoms. Each atom is described by a local Hilbert space spanned by the states $\{|g\rangle,|e\rangle\}$, referring respectively to the atomic ground and the highly-excited Rydberg states. The atoms are subject to a uniform laser drive with Rabi frequency $\Omega$ and detuning $\delta$, and they interact with one another via the Van der Waals potential $V(x)\approx r^{-6}$ at short distances. The resulting many-body Hamiltonian is +\begin{equation} +\hat{H} = -\Omega \sum_{\bm{r}}\hat{S}^x(\bm{r}) -\delta\sum_{\bm{r}}^N\hat{\Pi}(\bm{r}) +\frac{1}{2}\sum_{\bm{r},\bm{r^\prime}}V(\bm{r}-\bm{r^\prime})\hat{\Pi}(\bm{r}) \hat{\Pi}(\bm{r^\prime}) +\label{Eq::RydbergHamiltonian} +\end{equation} +where $\hat{\Pi}(\bm{r}) =|e\rangle\!\langle e|_{\bm{r}} $ is the projector onto the Rydberg state at position $\bm{r}$, $\hat{S}^x(\bm{r})=\frac{1}{2}\hat\sigma^x(\bm{r})$ are spin-$\frac{1}{2}$ operators, and $V(\bm{r}-\bm{r^\prime})=V_0/\|\bm{r}-\bm{r^\prime}\|^6$ is the Van der Waals potential between atoms at position $\bm{r}$ and $\bm{r^{\prime}}$. In the following, we assume $\Omega=1$ MHz. + +The phase diagram for the ground state of the Rydberg Hamiltonian is dictated by the mechanism of Rydberg blockade, a constraint that prevents two atoms at sufficiently small distances to be simultaneously excited to the Rydberg states. We can characterize the phase diagram in terms of the detuning $\delta$ and the interaction strength $V_0$. On the square lattice, several different orders have been detected by numerical simulations~\cite{Samajdar2020}. Here, we specifically focus on the $Z_2$ transition between a disordered phase at large and negative detuning, where all atoms are found in the ground state, and an ordered phase at large and positive detuning, where the system is found in one of the two symmetry-broken N\'eel states characterized by a checkerboard pattern in the atomic occupation number (Fig.~\ref{Fig::1}(a)). + +We perform numerical simulations of the ground state of Hamiltonian~(\ref{Eq::RydbergHamiltonian}) using the density matrix renormalization group (DMRG)~\cite{PhysRevLett.69.2863,PhysRevB.48.10345,SCHOLLWOCK201196} implemented using the software package ITensor~\cite{itensor}. We adopt a matrix product state (MPS) variational wavefunction $|\Psi\rangle$ with a snake-like geometry shown in Fig.~\ref{Fig::1}(b). We fix the interaction strength to $V_0=3$ MHz, and retain up to the third-nearest-neighbor interactions. For a several values of the detuning $\delta\in\{-5,5\}$ MHz, we run DMRG to find an approximation of the ground state, using a singular value decomposition cutoff of $10^{-10}$ and a target energy accuracy of $10^{-5}$. To certify convergence to the ground state, each run is repeated for different initialization of the starting MPS. + +We show the results of the simulations for a $8\times8$ array with open boundary conditions in Fig.~\ref{Fig::1}(c-f). We plot, as a function of the detuning, the ground state energy per site $E_0/N=\langle\Psi_0|\hat{H}|\Psi_0\rangle/N$, and the staggered magnetization $\langle\mathcal{N}\rangle=N^{-1}\sum_{\bm{r}}(-1)^{x+y}\langle\hat{S}^z(\bm{r})\rangle$, which can be used to detect N\'eel order. Whenever all atoms are in the ground state, $\langle\mathcal{N}\rangle\approx0$, while for an ordered state with a checkerboard pattern one has $\langle\mathcal{N}\rangle\approx0.5$. We also show the average occupation number in momentum space, +\begin{equation} +n(\bm{k})=\frac{1}{\sqrt{N}}\sum_{\bm{r}}e^{i\bm{k}\cdot\bm{r}}\langle\hat{n}(\bm{r})\rangle +\end{equation} +where $\hat{n}(\bm{r}) = \frac{1}{2}(1-2\hat{S}^z(\bm{r}))$. We observe a peak at $\linebreak\bm{k}=(\pi,\pi)$ for large detuning $\delta=4$ MHz (Fig.~\ref{Fig::1}(e)), and a featureless state at negative detuning (not shown). + +The two phases of the Rydberg atoms are separated by a second-order quantum phase transition at a critical point $\delta_c$. We can extract an approximation of $\delta_c$ by measuring the energy gap $\Delta = |E_0-E_1|$ between the ground state and the first excited state $|\Psi_1\rangle$. We compute $E_1$ by running DMRG on the Hamiltonian $\hat{H}^\prime=\hat{H}+\omega|\Psi\rangle\langle\Psi|$ where $\omega$ is an energy penalty. From the energy gap curve, we estimate the detuning where $\Delta\approx0$ to be $\delta_c\approx 1.3$ MHz. This approximate value will be sufficient for the purpose of this Article, though a more systematic scaling study with appropriate boundary conditions (to minimize finite-size effects) should be performed to accurately determine the critical point and critical exponents of the transition. + +Once we have solved for the ground states of the Rydberg Hamiltonian, the corresponding MPSs can be used to generate data to train the neural networks for the different applications. In this case, the data consists of projective measurements in the atomic occupation number basis $|\bm{\sigma}\rangle=|\sigma_1,\dots,\sigma_N\rangle$, where $\sigma_j=0$ and $\sigma_j=1$ refers respectively to the $j$-th atom being in the ground and Rydberg state. Given a wavefunction $|\Psi\rangle$, the probability to observe an atomic pattern $\bm{\sigma}$ following a measurement is simply given by the Born rule $P(\bm{\sigma})=|\langle\bm{\sigma}|\Psi\rangle|^2$. Because of the intrinsic one-dimensional geometry of an MPS, it is possible to efficiently sample the probability distribution $P(\bm{\sigma})$ by exploiting the chain rule of probabilities. Moreover, the sampling is exact in the sense that each sample is completely independent of one another~\cite{PhysRevB.85.165146}. %The sampling procedure consists of iteratively building one-site reduced density matrices conditional on the previous measurement outcomes~\cite{PhysRevB.85.165146}. + + + +%---------------------------------------------------------------------------------------- +%---------------------------------------------------------------------------------------- +% SUPERVISED LEARNING +%---------------------------------------------------------------------------------------- +%---------------------------------------------------------------------------------------- +%---------------------------------------------------------------------------------------- + + +\section{Learning a quantum phase transition} +\label{supervised} + +An important task in condensed matter and statistical physics is to characterize different phases of matter and the associated phase transitions between them. +Typically, phases of matter are described in terms of simple real-space patterns and their associated order parameters, which are theoretically understood using Landau symmetry-breaking paradigm~\cite{Xiao:803748}. While a wide array of theoretical and experimental tools to study interacting quantum systems have been constructed in relation to these patterns, there is an increasing set of states of matter whose theoretical and experimental understanding eludes the Landau symmetry-breaking paradigm. The characterization of these phases may rely on, e.g., out-of-equilibrium properties of the system as in the many-body localized phase~\cite{basko2006,RevModPhys.91.021001}, or on topological invariants in topological phases and spin liquids~\cite{Xiao:803748,savaryQuantumSpinLiquids2016, doi:10.1146/annurev-conmatphys-031218-013401}. + +Machine learning provides an alternative route to the characterization of phases of matter and their associated phase transitions in a semi-automated fashion without a direct use of manually specified real-space patterns and/or other signatures, provided that a sufficiently large training set is available. In its simplest form~\cite{carrasquilla2017nature}, given the existence of a classical or quantum phase transition between two phases in a physical system, one can use supervised learning to attempt to classify experimental or numerical snapshots of the phases of matter separated by the transition. This task can be achieved using most classification algorithms, e.g., those based on a neural network or a support vector machine~\cite{10.5555/1162264}, trained on snapshots of two phases of matter labelled according to the corresponding phase out of which the snapshot originated. Although here we only explore this simple strategy, we stress that machine learning approaches to studying phases and phase transitions have been significantly expanded and they no longer require the precise knowledge of the location of the critical point~\cite{evert2017nature,broecker2017b}, can be fully automatized, and can discover ordered phases~\cite{carrasquilla2017nature,leiwang2016,PhysRevE.96.022140}, topological phases~\cite{evert2017nature,rodriguez-nieva2019,PhysRevLett.125.127401}, and phases such as the many-body localized phase which is characterized by its dynamical properties~\cite{yi-ting2018,rao2018,PhysRevLett.120.257204}. + +The nature of the snapshots used to train the learning algorithms is vastly flexible, hence these strategies are of wide applicability, and can include numerically generated configurations visited during a classical or quantum Monte Carlo simulation of the physical system~\cite{carrasquilla2017nature,leiwang2016,evert2017nature,broecker2017,chng2017,PhysRevB.99.060404,PhysRevB.99.121104}, entanglement spectra~\cite{evert2017nature,yi-ting2018}, correlation matrices~\cite{PhysRevB.102.054512,PhysRevLett.125.170603}, tensors in an MPS~\cite{PhysRevLett.125.170603}, numerically generated projective measurements~\cite{berezutskii2020}, high-resolution real-space snapshots of complex many-body systems obtained with quantum gas microscopes for ultracold atoms~\cite{Bohrdt2018,PhysRevA.102.033326}, single-shot experimental momentum-space density images of ultracold quantum gases~\cite{Rem2018}, spectroscopic imaging scanning tunnelling microscopy data~\cite{Zhang_MLcuprates}, among many others. + +Below we explore learning a quantum phase in an array of interacting Rydberg atoms using projective measurements. As a classification algorithm, we make use a of a convolutional neural network~\cite{Goodfellow-et-al-2016} that readily takes advantage of the two-dimensional (2D) spatial arrangement of the the Rydberg atoms and their locality, as well as the approximate translation invariance of the system. + +\begin{figure}[t] +\noindent \centering{}\includegraphics[width=\columnwidth]{CNN.pdf} +\caption{A schematic representation of a convolutional neural network. The elements of the input $\mathsf{h}^{(0)}_{l,j,k}$ corresponds to the outcome of a projective measurement on the Rydberg system. The first operation is a convolutional layer with $ M_y \times M_x = 3 \times 3$ kernels with $I_{\text{out}} = 32$ output channels and $L_{\text{input}}=1$. This kernel is convolved with an input configuration with $N = 8 \times 8$ Rydberg atoms. Likewise, the second operation corresponds to a convolutional layer with $ M_y \times M_x = 3 \times 3$ kernels with $I_{\text{out}} = 32$ output channels and $L_{\text{input}}=32$. The output of the second convolutional layer is flattened and fed to an FC layer with a ReLU activation, followed by another FC layer with a softmax activation which produces the prediction outcome. } +\label{Fig::CNN} +\end{figure} + +\subsection{Convolutional neural networks and their training} +Convolutional neural networks (CNN) employ a mathematical operation called convolution to process information for data that has a natural grid-like topology~\cite{Goodfellow-et-al-2016}. A 2D convolutional layer implements the operation +\begin{align*} +\mathsf{h}^{(q)}_{i,j,k} & = F\left(\sum_{l,m_y,m_x} \mathsf{h}^{(q-1)}_{l,j+m_y,k+m_x} \mathsf{K}^{(q)}_{i,l,m_y,m_x} \right) \\ +&\coloneqq F\left( \mathsf{K}^{(q)} *\mathsf{h}^{(q-1)} \right) \nonumber %_{i,j,k} \nonumber +\end{align*} +where the trainable kernel $\mathsf{K}^{(q)}_{i,l,m_y,m_x}$ at layer $q$ specifies the connection strength between a unit in channel $i$ of the output and a unit in channel $l$ of the input, with a spatial offsets of $m_y$ rows (labeled y direction) and $m_x$ columns (labeled x direction) between the output and the input variables. The dimensions of the array $\mathsf{K}^{(q)}_{i,l,m,n}$ are $I_{\text{out}}$, $L_{\text{input}}$, $M_{y}$, $M_{x}$, which corresponds to the number of output channels, input channels, dimension of the filter along the vertical and horizontal directions, respectively. The activation at layer $q$ consists of elements $\mathsf{h}^{(q)}_{l,j,k}$, where $j$ and $k$ label vertical and horizontal directions, respectively, and $l$ specifies the channel. The activation units are labelled by $q$ where $q=0$ corresponds to the raw projective measurement data. Finally, the non-linear function $F(x)$, which in our examples is typically a rectified linear unit (ReLU) $F(x)=\text{max}(0,x)$, is applied element-wise to each of the components of its input. A convolutional neural network equipped with two convolutional layers is schematically shown in Fig.~\ref{Fig::CNN}. + + +Followed by the convolutional layers, a CNN typically processes information using sets of fully connected (FC) layers which implement a matrix-vector operation followed by a non-linearity $F$ as +\begin{equation} +\mathsf{h}^{(q)}_{i} = F\left(\sum_{l} \mathsf{h}^{(q-1)}_{l} \mathsf{K}^{(q)}_{i,l} + \mathsf{b}^{(q)}_{i}\right), +\end{equation} +where the trainable parameters of the FC layer are the kernel $\mathsf{K}^{(q)}_{i,l}$ and the bias vector $ \mathsf{b}^{(q)}_{i}$. +To feed the output of a convolutional layer $\mathsf{h}^{(q-1)}_{l,j,k}$ to an FC layer, the array is reshaped or ``flattened'' to $\mathsf{h^{\prime}}^{(q-1)}_{l}$ so that all the original components packed into a one-dimensional array with dimension $L_{\text{FC}}$. The last two layers of the CNN in Fig.~\ref{Fig::CNN} correspond to two fully connected layers with a ReLU and a softmax non-linearities, respectively. The softmax function $S$ is given by +\begin{equation} + \text{S}(\bm{v}) = \frac{\exp(\bm{v})}{\sum_i \exp(v_i)}. +\end{equation} +where $v_i$ are the components of a vector $\bm{v}$ and the $\text{exp}$ function acts element-wise on the components of the vector. We note that the input to the CNN and its trainable parameters are real, so that the outcome of the softmax layer can be interpreted as a probability distribution since $0 \le \text{S}(v_i)\le 1$ and $\sum_{i}\text{S}(v_i)=1$. + +Finally, we mention that we interpret our CNN as a model for the conditional probability of assigning a phase of matter $y=0,1$ to a projective measurement outcome $\bm{\sigma}=\mathsf{h}^{(0)}$, i.e. $P_{\bm{\theta}}(y|\bm{\sigma})$, where $\bm{\theta}$ encompasses all the trainable parameters of the CNN. The conditional is given by +\begin{widetext} +\begin{equation} + P_{\bm{\theta}}(y|\bm{\sigma}) = S\left( b^{(4)}_{y} + \sum_{m}\mathsf{K}^{(4)}_{y,m} F\left(b^{(3)}_{m} + \sum_{l} \mathsf{K}^{(3)}_{m,l}\,\text{Flatten}\left(F\left((\mathsf{K}^{(2)}*F\left(\mathsf{K}^{(1)}*\bm{\sigma} \right)\right)\right)_{l}\right) \right), + \label{Eq::CNNdist} +\end{equation} +\end{widetext} +where the function $\text{Flatten}()_{l}$ is the $l$-th component of a vector that arises from reshaping the incoming argument of the function to a one-dimensional array. + +%\subsection{Phase transition in the Rydberg array} + +To estimate the parameters of the CNN we use the maximum likelihood principle, where the parameters of a statistical model are selected by assigning high probability to the observed data. For a dataset with observations $\{\bm{\sigma}_n, y_n \}_{n=1}^{M}$, where $y_n=0,1$ label the phase of matter out of which a projective measurement $\bm{\sigma}_n$ was taken from, the likelihood assigned by the model to the dataset can be written as +\begin{equation} + p(\bm{y}|\bm{\theta}) = \prod_{n=1}^{M} + P_{\bm{\theta}}(y_n|\bm{\sigma}_n)^{y_n}(1-P_{\bm{\theta}}(y_n|\bm{\sigma}_n))^{1-y_n} +\end{equation} +where $\bm{y}=(y_1,...,y_{M})$. Instead of attempting to maximize the likelihood, it is convenient to define a loss function by taking the negative logarithm of the likelihood, which gives the cross-entropy +\begin{widetext} +\begin{equation} +E(\bm{\theta}) = -\ln( p(\bm{y}|\bm{\theta})) - = \sum_{n=1}^{M} \{ y_n \ln \left(P_{\bm{\theta}}(y_n|\bm{\sigma}_n))\right) + (1-y_n)\ln(1 - P_{\bm{\theta}}(y_n|\bm{\sigma}_n)) \}. +\label{Eq::NLL} +\end{equation} +\end{widetext} + +To train the model, we minimize $E(\bm{\theta})$ using gradient descent techniques~\cite{10.5555/1162264}. While it is possible to evaluate the gradients of $E(\bm{\theta})$ with respect to the parameters $\bm{\theta}$ in the CNN analytically using the chain rule, a more convenient and less error-prone approach is to use automatic differentiation (AD), which is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. A complete survery detailing AD can be found in Ref.~\cite{AD_2017}. + +In addition, instead of using the entire dataset in the calculation of $E(\bm{\theta})$ and its gradients, we use smaller batches of data of size $M_{\text{batch}}