Deep Learning of Quantum Many-Body Dynamics via Random Driving

Naeimeh Mohseni1,2, Thomas Fösel1,2, Lingzhen Guo1, Carlos Navarrete-Benlloch3,4,1, 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

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

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.

► BibTeX data

► References

[1] 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 (7152): 452–456, 2007. URL https:/​/​doi.org/​10.1038/​nature06011.
https:/​/​doi.org/​10.1038/​nature06011

[2] Matthew JS Beach, Anna Golubeva, and Roger G Melko. Machine learning vortices at the kosterlitz-thouless transition. Physical Review B, 97 (4): 045207, 2018. URL https:/​/​doi.org/​10.1103/​PhysRevB.97.045207.
https:/​/​doi.org/​10.1103/​PhysRevB.97.045207

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

[4] Giuseppe Carleo and Matthias Troyer. Solving the quantum many-body problem with artificial neural networks. Science, 355 (6325): 602–606, 2017. URL https:/​/​www.science.org/​doi/​10.1126/​science.aag2302.
https:/​/​www.science.org/​doi/​10.1126/​science.aag2302

[5] Juan Carrasquilla and Roger G Melko. Machine learning phases of matter. Nature Physics, 13 (5): 431–434, 2017. URL https:/​/​doi.org/​10.1038/​nphys4035.
https:/​/​doi.org/​10.1038/​nphys4035

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

[7] 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 (7382): 484–487, 2012. URL https:/​/​doi.org/​10.1038/​nature10748.
https:/​/​doi.org/​10.1038/​nature10748

[8] 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, Jul 2020. 10.1103/​PhysRevX.10.031002. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.10.031002.
https:/​/​doi.org/​10.1103/​PhysRevX.10.031002

[9] François Chollet et al. Keras. https:/​/​keras.io, 2015.
https:/​/​keras.io

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

[11] André Eckardt. Colloquium: Atomic quantum gases in periodically driven optical lattices. Rev. Mod. Phys., 89: 011004, Mar 2017. 10.1103/​RevModPhys.89.011004. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.89.011004.
https:/​/​doi.org/​10.1103/​RevModPhys.89.011004

[12] 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 (1): 011006, 2020.

[13] 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 (27): 6990–6992, 2014. URL https:/​/​doi.org/​10.1002/​anie.201404111.
https:/​/​doi.org/​10.1002/​anie.201404111

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

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

[16] 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, Jan 2018. 10.1103/​PhysRevX.8.011006. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevX.8.011006.
https:/​/​doi.org/​10.1103/​PhysRevX.8.011006

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

[18] 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, pages 6645–6649. Ieee, 2013. URL https:/​/​doi.org/​10.1109/​ICASSP.2013.6638947.
https:/​/​doi.org/​10.1109/​ICASSP.2013.6638947

[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 (6902): 51–54, 2002. URL https:/​/​doi.org/​10.1038/​nature00968.
https:/​/​doi.org/​10.1038/​nature00968

[20] Markus Philip Ludwig Heyl. Nonequilibrium phenomena in many-body quantum systems. PhD thesis, lmu, 2012. URL https:/​/​doi.org/​10.5282/​edoc.14583.
https:/​/​doi.org/​10.5282/​edoc.14583

[21] Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9 (8): 1735–1780, 1997. URL https:/​/​doi.org/​10.1162/​neco.1997.9.8.1735.
https:/​/​doi.org/​10.1162/​neco.1997.9.8.1735

[22] 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 (8): 1760–1772, 2012. URL https:/​/​doi.org/​10.48550/​arXiv.1110.0573.
https:/​/​doi.org/​10.48550/​arXiv.1110.0573

[23] Toshiya Kinoshita, Trevor Wenger, and David S Weiss. A quantum newton's cradle. Nature, 440 (7086): 900–903, 2006. URL https:/​/​doi.org/​10.1038/​nature04693.
https:/​/​doi.org/​10.1038/​nature04693

[24] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. nature, 521 (7553): 436–444, 2015. URL https:/​/​doi.org/​10.1038/​nature14539.
https:/​/​doi.org/​10.1038/​nature14539

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

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

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

[28] 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 (7608): 516–519, 2016. URL https:/​/​doi.org/​10.1038/​nature18318.
https:/​/​doi.org/​10.1038/​nature18318

[29] Glen Bigan Mbeng, Angelo Russomanno, and Giuseppe E Santoro. The quantum ising chain for beginners. arXiv preprint arXiv:2009.09208, 2020. URL https:/​/​doi.org/​10.48550/​arXiv.2009.09208.
https:/​/​doi.org/​10.48550/​arXiv.2009.09208
arXiv:2009.09208

[30] I. Medina and F. L. Semião. Pulse engineering for population control under dephasing and dissipation. Phys. Rev. A, 100: 012103, Jul 2019. 10.1103/​PhysRevA.100.012103. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.100.012103.
https:/​/​doi.org/​10.1103/​PhysRevA.100.012103

[31] 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, Jul 2013. 10.1103/​PhysRevLett.111.053003. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.111.053003.
https:/​/​doi.org/​10.1103/​PhysRevLett.111.053003

[32] 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 (6): 1221–1226, 2018. URL https:/​/​doi.org/​10.1073/​pnas.1714936115.
https:/​/​doi.org/​10.1073/​pnas.1714936115

[33] Kyle Mills, Pooya Ronagh, and Isaac Tamblyn. Finding the ground state of spin hamiltonians with reinforcement learning. Nature Machine Intelligence, 2 (9): 509–517, 2020. URL https:/​/​doi.org/​10.1038/​s42256-020-0226-x.
https:/​/​doi.org/​10.1038/​s42256-020-0226-x

[34] 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. URL https:/​/​doi.org/​10.48550/​arXiv.2109.08492.
https:/​/​doi.org/​10.48550/​arXiv.2109.08492
arXiv:2109.08492

[35] 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): 1–10, 2020. URL https:/​/​doi.org/​10.1038/​s41467-020-17835-9.
https:/​/​doi.org/​10.1038/​s41467-020-17835-9

[36] Michael A Nielsen. Neural networks and deep learning, volume 2018. Determination press San Francisco, CA, 2015.

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

[38] Vittorio Peano, Florian Sapper, and Florian Marquardt. Rapid exploration of topological band structures using deep learning. Physical Review X, 11 (2): 021052, 2021. URL https:/​/​doi.org/​10.1103/​PhysRevX.11.021052.
https:/​/​doi.org/​10.1103/​PhysRevX.11.021052

[39] 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, Apr 2011. 10.1103/​PhysRevLett.106.170501. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.106.170501.
https:/​/​doi.org/​10.1103/​PhysRevLett.106.170501

[40] 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 (7508): 198–201, 2014. URL https:/​/​doi.org/​10.1038/​nature13450.
https:/​/​doi.org/​10.1038/​nature13450

[41] Markus Schmitt and Markus Heyl. Quantum many-body dynamics in two dimensions with artificial neural networks. Phys. Rev. Lett., 125: 100503, Sep 2020. 10.1103/​PhysRevLett.125.100503. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.125.100503.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.100503

[42] Ulrich Schollwöck. The density-matrix renormalization group: a short introduction. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 369 (1946): 2643–2661, 2011. URL https:/​/​doi.org/​10.1098/​rsta.2010.0382.
https:/​/​doi.org/​10.1098/​rsta.2010.0382

[43] 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 (11): 738–743, 2013. URL https:/​/​doi.org/​10.1038/​nphys2750.
https:/​/​doi.org/​10.1038/​nphys2750

[44] 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. URL https:/​/​doi.org/​10.1038/​s41598-021-95562-x.
https:/​/​doi.org/​10.1038/​s41598-021-95562-x
arXiv:2012.04686

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

[46] Giacomo Torlai and Roger G. Melko. Neural decoder for topological codes. Phys. Rev. Lett., 119: 030501, Jul 2017. 10.1103/​PhysRevLett.119.030501. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.119.030501.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.030501

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

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

[49] L. M. K. Vandersypen and I. L. Chuang. Nmr techniques for quantum control and computation. Rev. Mod. Phys., 76: 1037–1069, Jan 2005. 10.1103/​RevModPhys.76.1037. URL https:/​/​link.aps.org/​doi/​10.1103/​RevModPhys.76.1037.
https:/​/​doi.org/​10.1103/​RevModPhys.76.1037

[50] Lei Wang. Discovering phase transitions with unsupervised learning. Phys. Rev. B, 94: 195105, Nov 2016. 10.1103/​PhysRevB.94.195105. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevB.94.195105.
https:/​/​doi.org/​10.1103/​PhysRevB.94.195105

[51] Sebastian J Wetzel. Unsupervised learning of phase transitions: From principal component analysis to variational autoencoders. Physical Review E, 96 (2): 022140, 2017. URL https:/​/​doi.org/​10.1103/​PhysRevE.96.022140.
https:/​/​doi.org/​10.1103/​PhysRevE.96.022140

[52] 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, Feb 2020. 10.1103/​PhysRevResearch.2.012039. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevResearch.2.012039.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.012039

Cited by

[1] Alex Blania, Sandro Herbig, Fabian Dechent, Evert van Nieuwenburg, and Florian Marquardt, "Deep learning of spatial densities in inhomogeneous correlated quantum systems", arXiv:2211.09050.

[2] 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.

[3] 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 2022-11-30 03:08:19). 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 2022-11-30 03:08:18).