Deep learning of many-body observables and quantum information scrambling

Naeimeh Mohseni; Junheng Shi; Tim Byrnes; Michael J. Hartmann

doi:10.22331/q-2024-07-18-1417

Deep learning of many-body observables and quantum information scrambling

Naeimeh Mohseni^1,2, Junheng Shi³, Tim Byrnes^3,4,5, and Michael J. Hartmann^1,2

¹Physics Department, Friedrich-Alexander Universität of Erlangen-Nuremberg, Staudtstr. 7, 91058 Erlangen, Germany
²Max-Planck-Institut für die Physik des Lichts, Staudtstrasse 2, 91058 Erlangen, Germany
³New York University Shanghai, 1555 Century Ave, Pudong, Shanghai 200122, China
⁴National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
⁵Department of Physics, New York University, New York, NY 10003, USA

Published:	2024-07-18, volume 8, page 1417
Eprint:	arXiv:2302.04621v2
Doi:	https://doi.org/10.22331/q-2024-07-18-1417
Citation:	Quantum 8, 1417 (2024).

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

Abstract

Machine learning has shown significant breakthroughs in quantum science, where in particular deep neural networks exhibited remarkable power in modeling quantum many-body systems. Here, we explore how the capacity of data-driven deep neural networks in learning the dynamics of physical observables is correlated with the scrambling of quantum information. We train a neural network to find a mapping from the parameters of a model to the evolution of observables in random quantum circuits for various regimes of quantum scrambling and test its $generalization$ and $extrapolation$ capabilities in applying it to unseen circuits. Our results show that a particular type of recurrent neural network is extremely powerful in generalizing its predictions within the system size and time window that it has been trained on for both, localized and scrambled regimes. These include regimes where classical learning approaches are known to fail in sampling from a representation of the full wave function. Moreover, the considered neural network succeeds in $extrapolating$ its predictions beyond the time window and system size that it has been trained on for models that show localization, but not in scrambled regimes.

Featured image: (a) The neural network learns a simple and efficient representation for the equations of motion when provided with the parameters of the model (shown with $\boldsymbol{\vec{\theta}}$) just by observing a subset of physical observables but without knowing the unitary operator that describes the model. (b) The propagation of information in a many-body system forms a light cone. For MBL models the radius of the light cone grows logarithmically while it propagates polynomially for scrambled models. Two qubits do not affect each other if one lies outside of the light cone of the other. The representation learned by the neural network is reliable beyond the system size and time window that it has been trained on where information scrambles slowly (logarithmically) and the dynamics is localized.

Popular summary

Machine learning has achieved significant breakthroughs in quantum science, with deep neural networks demonstrating remarkable power in simulating quantum many-body systems. In this work, we advance the understanding of data-driven classical learning algorithms in tackling complex, highly entangled, as well as localized quantum many-body dynamics. In particular, we show how the learning power of convolutional recurrent neural networks is connected to the propagation of quantum information in time and space. We explore this connection through an investigation of the dynamics generated by random quantum circuits, which enable us to interpolate between different regimes of localization and quantum scrambling. Our findings identify scenarios where data-driven approaches can efficiently describe the dynamics of many-body observables without requiring knowledge of the wave function. Moreover, we demonstrate that, for localized dynamics, the neural network can predict the dynamics of observables for larger system sizes and longer times than those, on which it was initially trained but fails in scrambled regimes.

► BibTeX data

@article{Mohseni2024deeplearningofmany,
  doi = {10.22331/q-2024-07-18-1417},
  url = {https://doi.org/10.22331/q-2024-07-18-1417},
  title = {Deep learning of many-body observables and quantum information scrambling},
  author = {Mohseni, Naeimeh and Shi, Junheng and Byrnes, Tim and Hartmann, Michael J.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1417},
  month = jul,
  year = {2024}
}

► 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] Leonardo Banchi, Edward Grant, Andrea Rocchetto, and Simone Severini. Modelling non-markovian quantum processes with recurrent neural networks. New Journal of Physics, 20 (12): 123030, 2018. 10.1088/1367-2630/aaf749.
https://doi.org/10.1088/1367-2630/aaf749

[3] Hannes Bernien, Sylvain Schwartz, Alexander Keesling, Harry Levine, Ahmed Omran, Hannes Pichler, Soonwon Choi, Alexander S Zibrov, Manuel Endres, Markus Greiner, et al. Probing many-body dynamics on a 51-atom quantum simulator. Nature, 551 (7682): 579–584, 2017. URL https://doi.org/10.1038/nature24622.
https://doi.org/10.1038/nature24622

[4] Sergio Boixo, Sergei V Isakov, Vadim N Smelyanskiy, Ryan Babbush, Nan Ding, Zhang Jiang, Michael J Bremner, John M Martinis, and Hartmut Neven. Characterizing quantum supremacy in near-term devices. Nature Physics, 14 (6): 595–600, 2018. URL https://doi.org/10.1038/s41567-018-0124-x.
https://doi.org/10.1038/s41567-018-0124-x

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

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

[7] J Ignacio Cirac, David Perez-Garcia, Norbert Schuch, and Frank Verstraete. Matrix product states and projected entangled pair states: Concepts, symmetries, theorems. Reviews of Modern Physics, 93 (4): 045003, 2021. URL https://doi.org/10.1103/RevModPhys.93.045003.
https://doi.org/10.1103/RevModPhys.93.045003

[8] Anna Dawid, Julian Arnold, Borja Requena, Alexander Gresch, Marcin Płodzień, Kaelan Donatella, Kim A Nicoli, Paolo Stornati, Rouven Koch, Miriam Büttner, et al. Modern applications of machine learning in quantum sciences. arXiv preprint arXiv:2204.04198, 2022. URL https://doi.org/10.48550/arXiv.2204.04198.
https://doi.org/10.48550/arXiv.2204.04198
arXiv:2204.04198

[9] Dong-Ling Deng, Xiaopeng Li, and S. Das Sarma. Quantum entanglement in neural network states. Phys. Rev. X, 7: 021021, May 2017. 10.1103/PhysRevX.7.021021. URL https://doi.org/10.1103/PhysRevX.7.021021.
https://doi.org/10.1103/PhysRevX.7.021021

[10] Jens Eisert, Mathis Friesdorf, and Christian Gogolin. Quantum many-body systems out of equilibrium. Nature Physics, 11 (2): 124–130, 2015. URL https://doi.org/10.1038/nphys3215.
https://doi.org/10.1038/nphys3215

[11] 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. URL https://doi.org/10.1103/PhysRevX.10.011006.
https://doi.org/10.1103/PhysRevX.10.011006

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

[13] Christian Gross and Immanuel Bloch. Quantum simulations with ultracold atoms in optical lattices. Science, 357 (6355): 995–1001, 2017. URL https://doi.org/10.1126/science.aal3837.
https://doi.org/10.1126/science.aal3837

[14] Eman Hamza, Robert Sims, and Günter Stolz. Dynamical localization in disordered quantum spin systems. Communications in Mathematical Physics, 315 (1): 215–239, 2012. URL https://doi.org/10.1007/s00220-012-1544-6.
https://doi.org/10.1007/s00220-012-1544-6

[15] Michael J. Hartmann and Giuseppe Carleo. Neural-network approach to dissipative quantum many-body dynamics. Phys. Rev. Lett., 122: 250502, Jun 2019. 10.1103/PhysRevLett.122.250502.
https://doi.org/10.1103/PhysRevLett.122.250502

[16] Luis E Herrera Rodriguez and Alexei A Kananenka. Convolutional neural networks for long time dissipative quantum dynamics. The Journal of Physical Chemistry Letters, 12 (9): 2476–2483, 2021. URL https://doi.org/10.1021/acs.jpclett.1c00079.
https://doi.org/10.1021/acs.jpclett.1c00079

[17] Marcel Hinsche, Marios Ioannou, Alexander Nietner, Jonas Haferkamp, Yihui Quek, Dominik Hangleiter, Jean-Pierre Seifert, Jens Eisert, and Ryan Sweke. Learnability of the output distributions of local quantum circuits. arXiv preprint arXiv:2110.05517, 2021. URL https://doi.org/10.48550/arXiv.2110.05517.
https://doi.org/10.48550/arXiv.2110.05517
arXiv:2110.05517

[18] Hsin-Yuan Huang, Sitan Chen, and John Preskill. Learning to predict arbitrary quantum processes. PRX Quantum, 4 (4): 040337, 2023. URL https://doi.org/10.1103/PRXQuantum.4.040337.
https://doi.org/10.1103/PRXQuantum.4.040337

[19] John Z Imbrie. On many-body localization for quantum spin chains. Journal of Statistical Physics, 163 (5): 998–1048, 2016. URL https://doi.org/10.1007/s10955-016-1508-x.
https://doi.org/10.1007/s10955-016-1508-x

[20] Zhih-Ahn Jia, Lu Wei, Yu-Chun Wu, Guang-Can Guo, and Guo-Ping Guo. Entanglement area law for shallow and deep quantum neural network states. New Journal of Physics, 22 (5): 053022, 2020. 10.1088/1367-2630/ab8262.
https://doi.org/10.1088/1367-2630/ab8262

[21] Hannah Lange, Anka Van de Walle, Atiye Abedinnia, and Annabelle Bohrdt. From architectures to applications: A review of neural quantum states. arXiv preprint arXiv:2402.09402, 2024. URL https://doi.org/10.48550/arXiv.2402.09402.
https://doi.org/10.48550/arXiv.2402.09402
arXiv:2402.09402

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

[23] Zi-Wen Liu, Seth Lloyd, Elton Yechao Zhu, and Huangjun Zhu. Generalized entanglement entropies of quantum designs. Phys. Rev. Lett., 120: 130502, Mar 2018. 10.1103/PhysRevLett.120.130502.
https://doi.org/10.1103/PhysRevLett.120.130502

[24] Paolo P Mazza, Dominik Zietlow, Federico Carollo, Sabine Andergassen, Georg Martius, and Igor Lesanovsky. Machine learning time-local generators of open quantum dynamics. Physical Review Research, 3 (2): 023084, 2021. URL https://doi.org/10.1103/PhysRevResearch.3.023084.
https://doi.org/10.1103/PhysRevResearch.3.023084

[25] et al. MD SAJID ANIS. Qiskit: An open-source framework for quantum computing, 2021.

[26] Matija Medvidović and Javier Robledo Moreno. Neural-network quantum states for many-body physics. arXiv preprint arXiv:2402.11014, 2024. URL https://doi.org/10.48550/arXiv.2402.11014.
https://doi.org/10.48550/arXiv.2402.11014
arXiv:2402.11014

[27] Naeimeh Mohseni, Thomas Fösel, Lingzhen Guo, Carlos Navarrete-Benlloch, and Florian Marquardt. Deep learning of quantum many-body dynamics via random driving. Quantum, 6: 714, 2022. URL https://doi.org/10.22331/q-2022-05-17-714.
https://doi.org/10.22331/q-2022-05-17-714

[28] Naeimeh Mohseni, Carlos Navarrete-Benlloch, Tim Byrnes, and Florian Marquardt. Deep recurrent networks predicting the gap evolution in adiabatic quantum computing. Quantum, 7: 1039, 2023. URL https://doi.org/10.22331/q-2023-06-12-1039.
https://doi.org/10.22331/q-2023-06-12-1039

[29] Emilia Morosan, Douglas Natelson, Andriy H Nevidomskyy, and Qimiao Si. Strongly correlated materials. Advanced Materials, 24 (36): 4896–4923, 2012. URL https://doi.org/10.1002/adma.201202018.
https://doi.org/10.1002/adma.201202018

[30] Adam Nahum, Sagar Vijay, and Jeongwan Haah. Operator spreading in random unitary circuits. Physical Review X, 8 (2): 021014, 2018. URL https://doi.org/10.1103/PhysRevX.8.021014.
https://doi.org/10.1103/PhysRevX.8.021014

[31] C. Navarrete-Benlloch, R. Garcés, N. Mohseni, and G. J. de Valcárcel. Floquet theory for temporal correlations and spectra in time-periodic open quantum systems: Application to squeezed parametric oscillation beyond the rotating-wave approximation. Phys. Rev. A, 103: 023713, Feb 2021. 10.1103/PhysRevA.103.023713. URL https://doi.org/10.1103/PhysRevA.103.023713.
https://doi.org/10.1103/PhysRevA.103.023713

[32] Murphy Yuezhen Niu, Andrew M Dai, Li Li, Augustus Odena, Zhengli Zhao, Vadim Smelyanskyi, Hartmut Neven, and Sergio Boixo. Learnability and complexity of quantum samples. arXiv preprint arXiv:2010.11983, 2020. URL https://doi.org/10.48550/arXiv.2010.11983.
https://doi.org/10.48550/arXiv.2010.11983
arXiv:2010.11983

[33] Manuel S Rudolph, Jacob Miller, Jing Chen, Atithi Acharya, and Alejandro Perdomo-Ortiz. Synergy between quantum circuits and tensor networks: Short-cutting the race to practical quantum advantage. arXiv preprint arXiv:2208.13673, 2022. URL https://doi.org/10.48550/arXiv.2208.13673.
https://doi.org/10.48550/arXiv.2208.13673
arXiv:2208.13673

[34] Philipp Schmidt, Florian Marquardt, and Naeimeh Mohseni. Transfer learning in predicting quantum many-body dynamics: from physical observables to entanglement entropy. arXiv preprint arXiv:2405.16254, 2024. URL https://doi.org/10.48550/arXiv.2405.16254.
https://doi.org/10.48550/arXiv.2405.16254
arXiv:2405.16254

[35] Markus Schmitt and Markus Heyl. Quantum dynamics in transverse-field ising models from classical networks. SciPost Phys, 4: 013, 2018.
https://doi.org/10.21468/SciPostPhys.4.2.013

[36] 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.
https://doi.org/10.1103/PhysRevLett.125.100503

[37] Maksym Serbyn, Z. Papić, and D. A. Abanin. Quantum quenches in the many-body localized phase. Phys. Rev. B, 90: 174302, Nov 2014. 10.1103/PhysRevB.90.174302. URL https://doi.org/10.1103/PhysRevB.90.174302.
https://doi.org/10.1103/PhysRevB.90.174302

[38] Xingjian Shi, Zhourong Chen, Hao Wang, Dit-Yan Yeung, Wai-Kin Wong, and Wang-chun Woo. Convolutional lstm network: A machine learning approach for precipitation nowcasting. Advances in neural information processing systems, 28, 2015.

[39] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017.

[40] SHI Xingjian, Zhourong Chen, Hao Wang, Dit-Yan Yeung, Wai-Kin Wong, and Wang-chun Woo. Convolutional lstm network: A machine learning approach for precipitation nowcasting. In Advances in neural information processing systems, pages 802–810, 2015.

[41] Yuan-Hang Zhang and Massimiliano Di Ventra. Transformer quantum state: A multipurpose model for quantum many-body problems. Physical Review B, 107 (7): 075147, 2023. 10.1103/PhysRevB.107.075147.
https://doi.org/10.1103/PhysRevB.107.075147

Cited by

[1] Philipp Schmidt, Florian Marquardt, and Naeimeh Mohseni, "Transfer learning in predicting quantum many-body dynamics: from physical observables to entanglement entropy", Quantum Science and Technology 10 2, 025038 (2025).

[2] Simone Cantori and Sebastiano Pilati, "Challenges and opportunities in the supervised learning of quantum circuit expectation values", Physical Review E 112 1, 015304 (2025).

[3] Anton Simen, Rodrigo Bloot, Otto M. Pires, and Erick G. Sperandio Nascimento, "Evolutionary-enhanced quantum supervised learning model", The Journal of Supercomputing 81 2, 388 (2025).

[4] Peter Sentz, Stanley Nicholson, Yujin Cho, Sohail Reddy, Brendan Keith, and Stefanie Günther, "Learning thermodynamic master equations for open quantum systems", Quantum 10, 2151 (2026).

[5] Jonas Blenninger, David Bucher, Giorgio Cortiana, Kumar Ghosh, Naeimeh Mohseni, Jonas Nüßlein, Corey O’Meara, Daniel Porawski, and Benedikt Wimmer, "Q-GRID: Quantum Optimization for the Future Energy Grid", KI - Künstliche Intelligenz 38 4, 339 (2024).

[6] Patrick Egenlauf, Iva Březinová, Sabine Andergassen, and Miriam Klopotek, "Capturing reduced-order quantum many-body dynamics out of equilibrium via neural ordinary differential equations", Machine Learning: Science and Technology 7 2, 025062 (2026).

[7] Ryui Kaneko, Masatoshi Imada, Yoshiyuki Kabashima, and Tomi Ohtsuki, "Forecasting long-time dynamics in quantum many-body systems by dynamic mode decomposition", Physical Review Research 7 1, 013085 (2025).

[8] I. D. Lazarev, Marek Narozniak, Tim Byrnes, and A. N. Pyrkov, "Hybrid quantum-classical unsupervised data clustering based on the self-organizing feature map", Physical Review A 111 1, 012416 (2025).

[9] Simone Cantori, Andrea Mari, David Vitali, and Sebastiano Pilati, "Synergy between noisy quantum computers and scalable classical deep learning for quantum error mitigation", EPJ Quantum Technology 11 1, 45 (2024).

[10] Zihao Qi, Yang Peng, and Christopher Earls, "Fourier Neural Operators for Time-Periodic Quantum Systems: Learning Floquet Hamiltonians, Observable Dynamics, and Operator Growth", PRX Quantum 7 2, 020340 (2026).

[11] Sohail Reddy, Stefanie Günther, and Yujin Cho, "Data-driven characterization of latent dynamics on quantum testbeds", AVS Quantum Science 6 3, 033803 (2024).

[12] Yue Yao, Yuqing Wang, Jixin Shi, Bo Liang, and Yixiang Shi, "A Novel Multi-Physics Coupled 10-Cell Stack Numerical Model for Microtubular Solid Oxide Fuel Cell Based on Single Rectangle Current Collector", Quantum Science and Technology 11 1, 79 (2026).

[13] Naeimeh Mohseni, Carlos Navarrete-Benlloch, Tim Byrnes, and Florian Marquardt, "Deep recurrent networks predicting the gap evolution in adiabatic quantum computing", Quantum 7, 1039 (2023).

[14] Naeimeh Mohseni, Thomas Morstyn, Corey O'Meara, David Bucher, Jonas Nüßlein, and Giorgio Cortiana, "Demonstrating Quantum Scaling Advantage in Approximate Optimization for Energy Coalition Formation with 100+ Agents", arXiv:2405.11917, (2024).

[15] Yan Zhu, Ya-Dong Wu, Qiushi Liu, Yuexuan Wang, and Giulio Chiribella, "Quantum Process Learning Through Neural Emulation", arXiv:2308.08815, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2026-07-15 13:07:20) and SAO/NASA ADS (last updated successfully 2026-07-15 13:07:21). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.