Deep Learning of Quantum Many-Body Dynamics via Random Driving

Naeimeh Mohseni1,2, Thomas Fösel1,2, Lingzhen Guo1, Carlos Navarrete-Benlloch1,3,4, and Florian Marquardt1,2

1Max-Planck-Institut für die Physik des Lichts, Staudtstrasse 2, 91058 Erlangen, Germany
2Physics Department, University of Erlangen-Nuremberg, Staudtstr. 5, 91058 Erlangen, Germany
3Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
4Shanghai Research Center for Quantum Sciences, Shanghai 201315, China

Abstract

Neural networks have emerged as a powerful way to approach many practical problems in quantum physics. In this work, we illustrate the power of deep learning to predict the dynamics of a quantum many-body system, where the training is $\textit{based purely on monitoring expectation values of observables under random driving}$. The trained recurrent network is able to produce accurate predictions for driving trajectories entirely different than those observed during training. As a proof of principle, here we train the network on numerical data generated from spin models, showing that it can learn the dynamics of observables of interest without needing information about the full quantum state. This allows our approach to be applied eventually to actual experimental data generated from a quantum many-body system that might be open, noisy, or disordered, without any need for a detailed understanding of the system. This scheme provides considerable speedup for rapid explorations and pulse optimization. Remarkably, we show the network is able to extrapolate the dynamics to times longer than those it has been trained on, as well as to the infinite-system-size limit.

One of the main outstanding challenges in quantum physics is an efficient treatment of nonequilibrium dynamics in quantum many-body systems. Direct simulations are constrained by the need to evolve the exponentially large many-body wave function, while ansatz solutions (including modern techniques like matrix product states) are typically restricted in their applicability. In this work, we introduce a novel approach to tackle this challenge, based on deep neural networks.

In contrast to the previous attempts to employ neural networks to represent variational wave functions, we entirely forego the need to deal with the quantum state itself. Rather, we show that we can teach a neural network to predict the nonequilibrium dynamics of a many-body quantum system, by having it observe the dynamics of a selected subset of degrees of freedom under random driving.

Being able to learn the dynamics by partial observations, without requiring information about the full state makes our scheme of high potential practical relevance. In particular, it immediately recommends itself to future applications in experiments where a full quantum-state tomography would be infeasible. In such cases, the network can be trained on experimental data without any knowledge of the underlying model. An independent benefit of our scheme is the significant speedup that could be used for certain tasks, e.g., for pulse engineering. The clear benefits arising from our scheme and the relative simplicity of the implementation make it a very promising approach for the prediction of quantum many-body dynamics.

► References

[1] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton, Deep learning,'' nature 521, 436–444 (2015).
https:/​/​doi.org/​10.1038/​nature14539

[2] Ian Goodfellow, Yoshua Bengio, Aaron Courville, and Yoshua Bengio, Deep learning, Vol. 1 (MIT press Cambridge, 2016).
https:/​/​dl.acm.org/​doi/​book/​10.5555/​3086952

[3] Lei Wang, Discovering phase transitions with unsupervised learning,'' Phys. Rev. B 94, 195105 (2016).
https:/​/​doi.org/​10.1103/​PhysRevB.94.195105

[4] Juan Carrasquilla and Roger G Melko, Machine learning phases of matter,'' Nature Physics 13, 431–434 (2017).
https:/​/​doi.org/​10.1038/​nphys4035

[5] Evert PL Van Nieuwenburg, Ye-Hua Liu, and Sebastian D Huber, Learning phase transitions by confusion,'' Nature Physics 13, 435–439 (2017).
https:/​/​doi.org/​10.1038/​nphys4037

[6] Sebastian J Wetzel, Unsupervised learning of phase transitions: From principal component analysis to variational autoencoders,'' Physical Review E 96, 022140 (2017).
https:/​/​doi.org/​10.1103/​PhysRevE.96.022140

[7] Matthew JS Beach, Anna Golubeva, and Roger G Melko, Machine learning vortices at the kosterlitz-thouless transition,'' Physical Review B 97, 045207 (2018).
https:/​/​doi.org/​10.1103/​PhysRevB.97.045207

[8] Giacomo Torlai, Guglielmo Mazzola, Juan Carrasquilla, Matthias Troyer, Roger Melko, and Giuseppe Carleo, Neural-network quantum state tomography,'' Nature Physics 14, 447–450 (2018).
https:/​/​doi.org/​10.1038/​s41567-018-0048-5

[9] Ivan Glasser, Nicola Pancotti, Moritz August, Ivan D. Rodriguez, and J. Ignacio Cirac, Neural-network quantum states, string-bond states, and chiral topological states,'' Phys. Rev. X 8, 011006 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.011006

[10] Giacomo Torlai and Roger G. Melko, Neural decoder for topological codes,'' Phys. Rev. Lett. 119, 030501 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.030501

[11] Thomas Fösel, Petru Tighineanu, Talitha Weiss, and Florian Marquardt, Reinforcement learning with neural networks for quantum feedback,'' Phys. Rev. X 8, 031084 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.031084

[12] Li Yang, Zhaoqi Leng, Guangyuan Yu, Ankit Patel, Wen-Jun Hu, and Han Pu, Deep learning-enhanced variational monte carlo method for quantum many-body physics,'' Phys. Rev. Research 2, 012039 (2020).
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.012039

[13] Kyle Mills, Pooya Ronagh, and Isaac Tamblyn, Finding the ground state of spin hamiltonians with reinforcement learning,'' Nature Machine Intelligence 2, 509–517 (2020).
https:/​/​doi.org/​10.1038/​s42256-020-0226-x

[14] Giuseppe Carleo and Matthias Troyer, Solving the quantum many-body problem with artificial neural networks,'' Science 355, 602–606 (2017).
https:/​/​doi.org/​10.1126/​science.aag2302

[15] Xun Gao and Lu-Ming Duan, Efficient representation of quantum many-body states with deep neural networks,'' Nature communications 8, 1–6 (2017).
https:/​/​doi.org/​10.1038/​s41467-017-00705-2

[16] Markus Schmitt and Markus Heyl, Quantum many-body dynamics in two dimensions with artificial neural networks,'' Phys. Rev. Lett. 125, 100503 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.125.100503

[17] Naeimeh Mohseni, Carlos Navarrete-Benlloch, Tim Byrnes, and Florian Marquardt, Deep recurrent networks predicting the gap evolution in adiabatic quantum computing,'' arXiv preprint arXiv:2109.08492 (2021).
https:/​/​doi.org/​10.48550/​arXiv.2109.08492
arXiv:2109.08492

[18] Philip Richerme, Zhe-Xuan Gong, Aaron Lee, Crystal Senko, Jacob Smith, Michael Foss-Feig, Spyridon Michalakis, Alexey V Gorshkov, and Christopher Monroe, Non-local propagation of correlations in quantum systems with long-range interactions,'' Nature 511, 198–201 (2014).
https:/​/​doi.org/​10.1038/​nature13450

[19] Markus Greiner, Olaf Mandel, Theodor W Hänsch, and Immanuel Bloch, Collapse and revival of the matter wave field of a bose–einstein condensate,'' Nature 419, 51–54 (2002).
https:/​/​doi.org/​10.1038/​nature00968

[20] F. Meinert, M. J. Mark, E. Kirilov, K. Lauber, P. Weinmann, A. J. Daley, and H.-C. Nägerl, Quantum quench in an atomic one-dimensional ising chain,'' Phys. Rev. Lett. 111, 053003 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.111.053003

[21] Toshiya Kinoshita, Trevor Wenger, and David S Weiss, A quantum newton's cradle,'' Nature 440, 900–903 (2006).
https:/​/​doi.org/​10.1038/​nature04693

[22] Esteban A Martinez, Christine A Muschik, Philipp Schindler, Daniel Nigg, Alexander Erhard, Markus Heyl, Philipp Hauke, Marcello Dalmonte, Thomas Monz, Peter Zoller, et al., Real-time dynamics of lattice gauge theories with a few-qubit quantum computer,'' Nature 534, 516–519 (2016).
https:/​/​doi.org/​10.1038/​nature18318

[23] Marc Cheneau, Peter Barmettler, Dario Poletti, Manuel Endres, Peter Schauß, Takeshi Fukuhara, Christian Gross, Immanuel Bloch, Corinna Kollath, and Stefan Kuhr, Light-cone-like spreading of correlations in a quantum many-body system,'' Nature 481, 484–487 (2012).
https:/​/​doi.org/​10.1038/​nature10748

[24] Pasquale Calabrese and John Cardy, Evolution of entanglement entropy in one-dimensional systems,'' Journal of Statistical Mechanics: Theory and Experiment 2005, P04010 (2005).
https:/​/​doi.org/​10.1088/​1742-5468/​2005/​04/​p04010

[25] Irene López-Gutiérrez and Christian B Mendl, Real time evolution with neural-network quantum states,'' arXiv preprint arXiv:1912.08831 (2019).
https:/​/​doi.org/​10.22331/​q-2022-01-20-627
arXiv:1912.08831

[26] David Poulin, Angie Qarry, Rolando Somma, and Frank Verstraete, Quantum simulation of time-dependent hamiltonians and the convenient illusion of hilbert space,'' Phys. Rev. Lett. 106, 170501 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.106.170501

[27] Julian Struck, Malte Weinberg, Christoph Ölschläger, Patrick Windpassinger, Juliette Simonet, Klaus Sengstock, Robert Höppner, Philipp Hauke, André Eckardt, Maciej Lewenstein, et al., Engineering ising-xy spin-models in a triangular lattice using tunable artificial gauge fields,'' Nature Physics 9, 738–743 (2013).
https:/​/​doi.org/​10.1038/​nphys2750

[28] André Eckardt, Colloquium: Atomic quantum gases in periodically driven optical lattices,'' Rev. Mod. Phys. 89, 011004 (2017).
https:/​/​doi.org/​10.1103/​RevModPhys.89.011004

[29] Mohammadali Foroozandeh, Ralph W Adams, Nicola J Meharry, Damien Jeannerat, Mathias Nilsson, and Gareth A Morris, Ultrahigh-resolution nmr spectroscopy,'' Angewandte Chemie International Edition 53, 6990–6992 (2014).
https:/​/​doi.org/​10.1002/​anie.201404111

[30] L. M. K. Vandersypen and I. L. Chuang, Nmr techniques for quantum control and computation,'' Rev. Mod. Phys. 76, 1037–1069 (2005).
https:/​/​doi.org/​10.1103/​RevModPhys.76.1037

[31] Marco Anderlini, Patricia J Lee, Benjamin L Brown, Jennifer Sebby-Strabley, William D Phillips, and James V Porto, Controlled exchange interaction between pairs of neutral atoms in an optical lattice,'' Nature 448, 452–456 (2007).
https:/​/​doi.org/​10.1038/​nature06011

[32] Román Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states,'' Annals of Physics 349, 117–158 (2014).
https:/​/​doi.org/​10.1016/​j.aop.2014.06.013

[33] Alex Graves, Abdel-rahman Mohamed, and Geoffrey Hinton, Speech recognition with deep recurrent neural networks,'' in 2013 IEEE international conference on acoustics, speech and signal processing (Ieee, 2013) pp. 6645–6649.
https:/​/​doi.org/​10.1109/​ICASSP.2013.6638947

[34] Ilya Sutskever, Oriol Vinyals, and Quoc V Le, Sequence to sequence learning with neural networks,'' Advances in neural information processing systems 27 (2014).
https:/​/​proceedings.neurips.cc/​paper/​2014/​file/​a14ac55a4f27472c5d894ec1c3c743d2-Paper.pdf

[35] Ulrich Schollwöck, The density-matrix renormalization group: a short introduction,'' Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 369, 2643–2661 (2011).
https:/​/​doi.org/​10.1098/​rsta.2010.0382

[36] Tom Struck, Javed Lindner, Arne Hollmann, Floyd Schauer, Andreas Schmidbauer, Dominique Bougeard, and Lars R Schreiber, Robust and fast post-processing of single-shot spin qubit detection events with a neural network,'' arXiv preprint arXiv:2012.04686 (2020).
https:/​/​doi.org/​10.1038/​s41598-021-95562-x
arXiv:2012.04686

[37] Emmanuel Flurin, Leigh S Martin, Shay Hacohen-Gourgy, and Irfan Siddiqi, Using a recurrent neural network to reconstruct quantum dynamics of a superconducting qubit from physical observations,'' Physical Review X 10, 011006 (2020).
https:/​/​doi.org/​10.1103/​PhysRevX.10.011006

[38] Joonhee Choi, Hengyun Zhou, Helena S. Knowles, Renate Landig, Soonwon Choi, and Mikhail D. Lukin, Robust dynamic hamiltonian engineering of many-body spin systems,'' Phys. Rev. X 10, 031002 (2020).
https:/​/​doi.org/​10.1103/​PhysRevX.10.031002

[39] I. Medina and F. L. Semião, Pulse engineering for population control under dephasing and dissipation,'' Phys. Rev. A 100, 012103 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.100.012103

[40] Alexey A Melnikov, Hendrik Poulsen Nautrup, Mario Krenn, Vedran Dunjko, Markus Tiersch, Anton Zeilinger, and Hans J Briegel, Active learning machine learns to create new quantum experiments,'' Proceedings of the National Academy of Sciences 115, 1221–1226 (2018).
https:/​/​doi.org/​10.1073/​pnas.1714936115

[41] H Moon, DT Lennon, J Kirkpatrick, NM van Esbroeck, LC Camenzind, Liuqi Yu, F Vigneau, DM Zumbühl, G Andrew D Briggs, MA Osborne, et al., Machine learning enables completely automatic tuning of a quantum device faster than human experts,'' Nature communications 11, 1–10 (2020).
https:/​/​doi.org/​10.1038/​s41467-020-17835-9

[42] Vittorio Peano, Florian Sapper, and Florian Marquardt, Rapid exploration of topological band structures using deep learning,'' Physical Review X 11, 021052 (2021).
https:/​/​doi.org/​10.1103/​PhysRevX.11.021052

[43] Yang Liu, Jingfa Li, Shuyu Sun, and Bo Yu, Advances in gaussian random field generation: a review,'' Computational Geosciences , 1–37 (2019).
https:/​/​doi.org/​10.1007/​s10596-019-09867-y

[44] Sepp Hochreiter and Jürgen Schmidhuber, Long short-term memory,'' Neural computation 9, 1735–1780 (1997).
https:/​/​doi.org/​10.1162/​neco.1997.9.8.1735

[45] Stefanie Czischek, Martin Gärttner, and Thomas Gasenzer, Quenches near ising quantum criticality as a challenge for artificial neural networks,'' Phys. Rev. B 98, 024311 (2018).
https:/​/​doi.org/​10.1103/​PhysRevB.98.024311

[46] Markus Philip Ludwig Heyl, Nonequilibrium phenomena in many-body quantum systems, Ph.D. thesis, lmu (2012).
https:/​/​doi.org/​10.5282/​edoc.14583

[47] Bikas K Chakrabarti, Amit Dutta, and Parongama Sen, Quantum Ising phases and transitions in transverse Ising models, Vol. 41 (Springer Science & Business Media, 2008).
https:/​/​doi.org/​10.1007/​978-3-642-33039-1

[48] Leonard Mandel and Emil Wolf, Optical coherence and quantum optics (Cambridge university press, 1995).
https:/​/​doi.org/​10.1017/​CBO9781139644105

[49] Glen Bigan Mbeng, Angelo Russomanno, and Giuseppe E Santoro, The quantum ising chain for beginners,'' arXiv preprint arXiv:2009.09208.
https:/​/​doi.org/​10.48550/​arXiv.2009.09208
arXiv:2009.09208

[50] Michael A Nielsen, Neural networks and deep learning, Vol. 2018 (Determination press San Francisco, CA, 2015).

[51] François Chollet et al., Keras,'' https:/​/​keras.io (2015).
https:/​/​keras.io

[52] J Robert Johansson, Paul D Nation, and Franco Nori, Qutip: An open-source python framework for the dynamics of open quantum systems,'' Computer Physics Communications 183, 1760–1772 (2012).
https:/​/​doi.org/​10.48550/​arXiv.1110.0573

Cited by

[1] Yaroslav Kharkov, Oles Shtanko, Alireza Seif, Przemyslaw Bienias, Mathias Van Regemortel, Mohammad Hafezi, and Alexey V. Gorshkov, "Discovering hydrodynamic equations of many-body quantum systems", arXiv:2111.02385.

[2] Naeimeh Mohseni, Carlos Navarrete-Benlloch, Tim Byrnes, and Florian Marquardt, "Deep recurrent networks predicting the gap evolution in adiabatic quantum computing", arXiv:2109.08492.

The above citations are from SAO/NASA ADS (last updated successfully 2022-07-05 06:54:37). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2022-07-05 06:54:35).