Real time evolution with neural-network quantum states

Irene López Gutiérrez and Christian B. Mendl

Technische Universität München, Department of Informatics and Institute for Advanced Study, Boltzmannstraße 3, 85748 Garching, Germany
Technische Universität Dresden, Institute of Scientific Computing, Zellescher Weg 12-14, 01069 Dresden, Germany

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

Abstract

A promising application of neural-network quantum states is to describe the time dynamics of many-body quantum systems. To realize this idea, we employ neural-network quantum states to approximate the implicit midpoint rule method, which preserves the symplectic form of Hamiltonian dynamics. We ensure that our complex-valued neural networks are holomorphic functions, and exploit this property to efficiently compute gradients. Application to the transverse-field Ising model on a one- and two-dimensional lattice exhibits an accuracy comparable to the stochastic configuration method proposed in [Carleo and Troyer, Science 355, 602-606 (2017)], but does not require computing the (pseudo-)inverse of a matrix.

Numerical simulation plays an important role for the analysis and understanding of quantum systems. An inherent difficulty stems from the exponential growth of the system's Hilbert space dimension with respect to its size. In particular, the long time dynamics of a quantum system, and the associated entanglement growth, precludes the efficient representation of the statevector by tensor network methods in general.

In this work, we approach the problem using machine learning methods, as proposed in [Carleo and Troyer, Science 355, 602-606 (2017)]. The statevector is approximated by a neural network, and the time evolution of the system is realized by changes in the network parameters. Our main contribution is a translation of well-established numerical integration methods to the optimization of the network weights. We construct a cost function based on the implicit midpoint method as a concrete example, and minimize it with gradient descent. Compared with the previously proposed stochastic reconfiguration method, our approach does not require a possibly ill-conditioned matrix inversion.

The paper describes the mathematical details of the optimization and provides numerical examples of the time dynamics governed by the Ising model, on a 1D lattice with 20 sites and a small 2D lattice. We find the accuracies achieved by our method to be on par with stochastic reconfiguration. The remaining error in the simulations points towards limits of the expressibility of the neural network ansatze themselves, which could be improved in future works.

► BibTeX data

► References

[1] V. Alba and P. Calabrese. Entanglement and thermodynamics after a quantum quench in integrable systems. PNAS, 114: 7947–7951, 2017. 10.1073/​pnas.1703516114.
https:/​/​doi.org/​10.1073/​pnas.1703516114

[2] H. W. J. Blöte and Y. Deng. Cluster Monte Carlo simulation of the transverse Ising model. Phys. Rev. E, 66: 066110, 2002. 10.1103/​PhysRevE.66.066110.
https:/​/​doi.org/​10.1103/​PhysRevE.66.066110

[3] A. Borin and D. A. Abanin. Approximating power of machine-learning ansatz for quantum many-body states. Phys. Rev. B, 101, 2020. 10.1103/​PhysRevB.101.195141.
https:/​/​doi.org/​10.1103/​PhysRevB.101.195141

[4] P. Calabrese and J. Cardy. Evolution of entanglement entropy in one-dimensional systems. J. Stat. Mech.: Theory Exp., 2005: P04010, 2005. 10.1088/​1742-5468/​2005/​04/​p04010.
https:/​/​doi.org/​10.1088/​1742-5468/​2005/​04/​p04010

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

[6] G. Carleo, F. Becca, M. Schiró, and M. Fabrizio. Localization and glassy dynamics of many-body quantum systems. Sci. Rep., 2: 243, 2012. 10.1038/​srep00243.
https:/​/​doi.org/​10.1038/​srep00243

[7] G. Carleo, F. Becca, L. Sanchez-Palencia, S. Sorella, and M. Fabrizio. Light-cone effect and supersonic correlations in one- and two-dimensional bosonic superfluids. Phys. Rev. A, 89: 031602, 2014. 10.1103/​PhysRevA.89.031602.
https:/​/​doi.org/​10.1103/​PhysRevA.89.031602

[8] S. R. Clark. Unifying neural-network quantum states and correlator product states via tensor networks. J. Phys. A Math. Theor., 51: 135301, 2018. 10.1088/​1751-8121/​aaaaf2.
https:/​/​doi.org/​10.1088/​1751-8121/​aaaaf2

[9] S. Czischek, M. Gärttner, and T. Gasenzer. Quenches near Ising quantum criticality as a challenge for artificial neural networks. Phys. Rev. B, 98: 024311, 2018. 10.1103/​PhysRevB.98.024311.
https:/​/​doi.org/​10.1103/​PhysRevB.98.024311

[10] A. J. Daley, C. Kollath, U. Schollwöck, and G. Vidal. Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces. J. Stat. Mech. Theory Exp., 2004: P04005, 2004. 10.1088/​1742-5468/​2004/​04/​p04005.
https:/​/​doi.org/​10.1088/​1742-5468/​2004/​04/​p04005

[11] D. Deng, X. Li, and S. Das Sarma. Quantum entanglement in neural network states. Phys. Rev. X, 7: 021021, 2017. 10.1103/​PhysRevX.7.021021.
https:/​/​doi.org/​10.1103/​PhysRevX.7.021021

[12] X. Gao and L.-M. Duan. Efficient representation of quantum many-body states with deep neural networks. Nat. Commun., 8: 662, 2017. 10.1038/​s41467-017-00705-2.
https:/​/​doi.org/​10.1038/​s41467-017-00705-2

[13] I. Glasser, N. Pancotti, M. August, I. D. Rodriguez, and J. I. Cirac. Neural-network quantum states, string-bond states, and chiral topological states. Phys. Rev. X, 8: 011006, 2018. 10.1103/​PhysRevX.8.011006.
https:/​/​doi.org/​10.1103/​PhysRevX.8.011006

[14] E. Hairer, C. Lubich, and G. Wanner. Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer-Verlag Berlin Heidelberg, 2006. 10.1007/​3-540-30666-8.
https:/​/​doi.org/​10.1007/​3-540-30666-8

[15] A. Hirose. Complex-Valued Neural Networks. Springer-Verlag Berlin Heidelberg, 2012. 10.1007/​978-3-642-27632-3.
https:/​/​doi.org/​10.1007/​978-3-642-27632-3

[16] M. Hochbruck and C. Lubich. Error analysis of Krylov methods in a nutshell. SIAM J. Sci. Comput., 19 (2): 695–701, 1998. 10.1137/​S1064827595290450.
https:/​/​doi.org/​10.1137/​S1064827595290450

[17] R. Kaubruegger, L. Pastori, and J. C. Budich. Chiral topological phases from artificial neural networks. Phys. Rev. B, 97: 195136, 2018. 10.1103/​PhysRevB.97.195136.
https:/​/​doi.org/​10.1103/​PhysRevB.97.195136

[18] D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In 3rd International Conference for Learning Representations, San Diego, 2015.

[19] Y. Levine, O. Sharir, N. Cohen, and A. Shashua. Quantum entanglement in deep learning architectures. Phys. Rev. Lett., 122: 065301, 2019. 10.1103/​PhysRevLett.122.065301.
https:/​/​doi.org/​10.1103/​PhysRevLett.122.065301

[20] J. Liesen and P. Tichý. Convergence analysis of Krylov subspace methods. GAMM-Mitteilungen, 27: 153–173, 2004. 10.1002/​gamm.201490008.
https:/​/​doi.org/​10.1002/​gamm.201490008

[21] Y. Nomura, A. S. Darmawan, Y. Yamaji, and M. Imada. Restricted Boltzmann machine learning for solving strongly correlated quantum systems. Phys. Rev. B, 96: 205152, 2017. 10.1103/​PhysRevB.96.205152.
https:/​/​doi.org/​10.1103/​PhysRevB.96.205152

[22] L. Pastori, R. Kaubruegger, and J. C. Budich. Generalized transfer matrix states from artificial neural networks. Phys. Rev. B, 99: 165123, 2019. 10.1103/​PhysRevB.99.165123.
https:/​/​doi.org/​10.1103/​PhysRevB.99.165123

[23] D. Poulin, A. Qarry, R. Somma, and F. Verstraete. Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space. Phys. Rev. Lett., 106: 170501, 2011. 10.1103/​PhysRevLett.106.170501.
https:/​/​doi.org/​10.1103/​PhysRevLett.106.170501

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

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

[26] U. Schollwöck. The density-matrix renormalization group. Rev. Mod. Phys., 77: 259–315, 2005. 10.1103/​RevModPhys.77.259.
https:/​/​doi.org/​10.1103/​RevModPhys.77.259

[27] U. Schollwöck. The density-matrix renormalization group in the age of matrix product states. Ann. Phys., 326: 96–192, 2011. 10.1016/​j.aop.2010.09.012.
https:/​/​doi.org/​10.1016/​j.aop.2010.09.012

[28] A. Shrestha and A. Mahmood. Review of deep learning algorithms and architectures. IEEE Access, 7: 53040–53065, 2019. 10.1109/​ACCESS.2019.2912200.
https:/​/​doi.org/​10.1109/​ACCESS.2019.2912200

[29] S. Sorella. Generalized Lanczos algorithm for variational quantum Monte Carlo. Phys. Rev. B, 64: 024512, 2001. 10.1103/​PhysRevB.64.024512.
https:/​/​doi.org/​10.1103/​PhysRevB.64.024512

[30] S. Suzuki, J. Inoue, and B. K. Chakrabarti. Quantum Ising Phases and Transitions in Transverse Ising Models. Springer, Berlin, Heidelberg, 2013. ISBN 978-3-642-33039-1. 10.1007/​978-3-642-33039-1.
https:/​/​doi.org/​10.1007/​978-3-642-33039-1

[31] C. Trabelsi, O. Bilaniuk, Y. Zhang, D. Serdyuk, S. Subramanian, J. F. Santos, S. Mehri, N. Rostamzadeh, Y. Bengio, and C. J. Pal. Deep complex networks. In International Conference on Learning Representations, 2018. URL https:/​/​openreview.net/​forum?id=H1T2hmZAb.
https:/​/​openreview.net/​forum?id=H1T2hmZAb

[32] V. N. Vapnik. An overview of statistical learning theory. IEEE Trans. Neural Netw., 10: 988–999, 1999. 10.1109/​72.788640.
https:/​/​doi.org/​10.1109/​72.788640

[33] G. Vidal. Efficient simulation of one-dimensional quantum many-body systems. Phys. Rev. Lett., 93: 040502, 2004. 10.1103/​PhysRevLett.93.040502.
https:/​/​doi.org/​10.1103/​PhysRevLett.93.040502

[34] S. R. White and A. E. Feiguin. Real-time evolution using the density matrix renormalization group. Phys. Rev. Lett., 93: 076401, 2004. 10.1103/​PhysRevLett.93.076401.
https:/​/​doi.org/​10.1103/​PhysRevLett.93.076401

Cited by

[1] Agnes Valenti, Guliuxin Jin, Julian Léonard, Sebastian D. Huber, and Eliska Greplova, "Scalable Hamiltonian learning for large-scale out-of-equilibrium quantum dynamics", Physical Review A 105 2, 023302 (2022).

[2] John M. Martyn, Khadijeh Najafi, and Di Luo, "Variational Neural-Network Ansatz for Continuum Quantum Field Theory", Physical Review Letters 131 8, 081601 (2023).

[3] Damian Hofmann, Giammarco Fabiani, Johan Mentink, Giuseppe Carleo, and Michael Sentef, "Role of stochastic noise and generalization error in the time propagation of neural-network quantum states", SciPost Physics 12 5, 165 (2022).

[4] Wenxuan Zhang, Xiansong Xu, Zheyu Wu, Vinitha Balachandran, and Dario Poletti, "Ground state search by local and sequential updates of neural network quantum states", Physical Review B 107 16, 165149 (2023).

[5] Kevin Slagle, "Quantum Gauge Networks: A New Kind of Tensor Network", Quantum 7, 1113 (2023).

[6] Eimantas Ledinauskas and Egidijus Anisimovas, "Scalable imaginary time evolution with neural network quantum states", SciPost Physics 15 6, 229 (2023).

[7] Markus Schmitt and Moritz Reh, "jVMC: Versatile and performant variational Monte Carlo leveraging automated differentiation and GPU acceleration", SciPost Physics Codebases 2 (2022).

[8] Roger G. Melko and Juan Carrasquilla, "Language models for quantum simulation", Nature Computational Science 4 1, 11 (2024).

[9] Han-Qing Shi and Hai-Qing Zhang, "Learning topological defects formation with neural networks in a quantum phase transition", Communications in Theoretical Physics 76 5, 055101 (2024).

[10] Vladimir Vargas-Calderón, Herbert Vinck-Posada, and Fabio A. González, "An empirical study of quantum dynamics as a ground state problem with neural quantum states", Quantum Information Processing 22 4, 165 (2023).

[11] Filippo Vicentini, Damian Hofmann, Attila Szabó, Dian Wu, Christopher Roth, Clemens Giuliani, Gabriel Pescia, Jannes Nys, Vladimir Vargas-Calderón, Nikita Astrakhantsev, and Giuseppe Carleo, "NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems", SciPost Physics Codebases 7 (2022).

[12] Pascal M. Vecsei, Christian Flindt, and Jose L. Lado, "Lee-Yang theory of quantum phase transitions with neural network quantum states", Physical Review Research 5 3, 033116 (2023).

[13] Zheyu Wu, Remmy Zen, Heitor P. Casagrande, Dario Poletti, and Stéphane Bressan, "Supervised training of neural-network quantum states for the next-nearest neighbor Ising model", Computer Physics Communications 300, 109169 (2024).

[14] Alexander Zaytsev, Darya Zaytseva, Sergey Zaytsev, Lorenzo Ugo Ancarani, Yury Popov, and Konstantin Kouzakov, "Parabolic wave packets for time propagation of atomic hydrogen in an electric field of short laser pulses", The European Physical Journal Plus 139 2, 199 (2024).

[15] Moritz Reh, Markus Schmitt, and Martin Gärttner, "Optimizing design choices for neural quantum states", Physical Review B 107 19, 195115 (2023).

[16] Di Luo, Zhuo Chen, Kaiwen Hu, Zhizhen Zhao, Vera Mikyoung Hur, and Bryan K. Clark, "Gauge-invariant and anyonic-symmetric autoregressive neural network for quantum lattice models", Physical Review Research 5 1, 013216 (2023).

[17] Aaron Sander, Lukas Burgholzer, and Robert Wille, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 283 (2023) ISBN:979-8-3503-4323-6.

[18] Kevin Slagle and John Preskill, "Emergent quantum mechanics at the boundary of a local classical lattice model", Physical Review A 108 1, 012217 (2023).

[19] 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).

[20] Ryui Kaneko and Ippei Danshita, "Dynamics of correlation spreading in low-dimensional transverse-field Ising models", Physical Review A 108 2, 023301 (2023).

[21] Ilaria Gianani and Claudia Benedetti, "Multiparameter estimation of continuous-time quantum walk Hamiltonians through machine learning", AVS Quantum Science 5 1, 014405 (2023).

[22] Tianchen Zhao, Chuhao Sun, Asaf Cohen, James Stokes, and Shravan Veerapaneni, "Quantum-inspired variational algorithms for partial differential equations: application to financial derivative pricing", Quantitative Finance 24 1, 1 (2024).

[23] Kaelan Donatella, Zakari Denis, Alexandre Le Boité, and Cristiano Ciuti, "Dynamics with autoregressive neural quantum states: Application to critical quench dynamics", Physical Review A 108 2, 022210 (2023).

[24] Alessandro Sinibaldi, Clemens Giuliani, Giuseppe Carleo, and Filippo Vicentini, "Unbiasing time-dependent Variational Monte Carlo by projected quantum evolution", Quantum 7, 1131 (2023).

[25] Markus Schmitt and Markus Heyl, "Quantum Many-Body Dynamics in Two Dimensions with Artificial Neural Networks", Physical Review Letters 125 10, 100503 (2020).

[26] Christopher Orthodoxou, Amelle Zaïr, and George H. Booth, "High harmonic generation in two-dimensional Mott insulators", npj Quantum Materials 6, 76 (2021).

[27] Sheng-Hsuan Lin and Frank Pollmann, "Scaling of Neural-Network Quantum States for Time Evolution", Physica Status Solidi B Basic Research 259 5, 2100172 (2022).

[28] Di Luo, Zhuo Chen, Juan Carrasquilla, and Bryan K. Clark, "Autoregressive Neural Network for Simulating Open Quantum Systems via a Probabilistic Formulation", Physical Review Letters 128 9, 090501 (2022).

[29] Benedikt Kloss, David Reichman, and Yevgeny Bar Lev, "Studying dynamics in two-dimensional quantum lattices using tree tensor network states", SciPost Physics 9 5, 070 (2020).

[30] Roberto Verdel, Markus Schmitt, Yi-Ping Huang, Petr Karpov, and Markus Heyl, "Variational classical networks for dynamics in interacting quantum matter", Physical Review B 103 16, 165103 (2021).

[31] Di Luo, Zhuo Chen, Juan Carrasquilla, and Bryan K. Clark, "Autoregressive Neural Network for Simulating Open Quantum Systems via a Probabilistic Formulation", arXiv:2009.05580, (2020).

[32] Di Luo, Aidan P. Reddy, Trithep Devakul, and Liang Fu, "Artificial intelligence for artificial materials: moiré atom", arXiv:2303.08162, (2023).

[33] Zhuo Chen, Di Luo, Kaiwen Hu, and Bryan K. Clark, "Simulating 2+1D Lattice Quantum Electrodynamics at Finite Density with Neural Flow Wavefunctions", arXiv:2212.06835, (2022).

[34] David R. Vivas, Javier Madroñero, Victor Bucheli, Luis O. Gómez, and John H. Reina, "Neural-Network Quantum States: A Systematic Review", arXiv:2204.12966, (2022).

[35] Di Luo, David D. Dai, and Liang Fu, "Pairing-based graph neural network for simulating quantum materials", arXiv:2311.02143, (2023).

[36] Chu Guo and Dario Poletti, "Scheme for automatic differentiation of complex loss functions with applications in quantum physics", Physical Review E 103 1, 013309 (2021).

[37] Chee Kong Lee, Pranay Patil, Shengyu Zhang, and Chang Yu Hsieh, "Neural-network variational quantum algorithm for simulating many-body dynamics", Physical Review Research 3 2, 023095 (2021).

[38] Di Luo and James Halverson, "Infinite neural network quantum states: entanglement and training dynamics", Machine Learning: Science and Technology 4 2, 025038 (2023).

[39] Benedikt Fauseweh and Jian-Xin Zhu, "Laser pulse driven control of charge and spin order in the two-dimensional Kondo lattice", Physical Review B 102 16, 165128 (2020).

[40] Markus Schmitt and Moritz Reh, "jVMC: Versatile and performant variational Monte Carlo leveraging automated differentiation and GPU acceleration", arXiv:2108.03409, (2021).

[41] Saeed S. Jahromi and Roman Orus, "Variational Tensor Neural Networks for Deep Learning", arXiv:2211.14657, (2022).

[42] Kevin Slagle, "Testing Quantum Mechanics using Noisy Quantum Computers", arXiv:2108.02201, (2021).

[43] Stefano De Nicola, "Importance sampling scheme for the stochastic simulation of quantum spin dynamics", SciPost Physics 11 3, 048 (2021).

[44] I. Meyerov, A. Liniov, M. Ivanchenko, and S. Denisov, "Simulating quantum dynamics: Evolution of algorithms in the HPC context", arXiv:2005.04681, (2020).

[45] Rouven Koch and Jose L. Lado, "Neural network enhanced hybrid quantum many-body dynamical distributions", Physical Review Research 3 3, 033102 (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2024-04-19 01:24:23) and SAO/NASA ADS (last updated successfully 2024-04-19 01:24:24). The list may be incomplete as not all publishers provide suitable and complete citation data.