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.
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.
 V. Alba and P. Calabrese. Entanglement and thermodynamics after a quantum quench in integrable systems. PNAS, 114: 7947–7951, 2017. 10.1073/pnas.1703516114.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In 3rd International Conference for Learning Representations, San Diego, 2015.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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).
 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).
 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).
 Markus Schmitt and Moritz Reh, "jVMC: Versatile and performant variational Monte Carlo leveraging automated differentiation and GPU acceleration", SciPost Physics Codebases 2 (2022).
 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).
 Markus Schmitt and Markus Heyl, "Quantum Many-Body Dynamics in Two Dimensions with Artificial Neural Networks", Physical Review Letters 125 10, 100503 (2020).
 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).
 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).
 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).
 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).
 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).
 Christopher Orthodoxou, Amelle Zaïr, and George H. Booth, "High harmonic generation in two-dimensional Mott insulators", npj Quantum Materials 6, 76 (2021).
 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).
 Stefano De Nicola, "Importance sampling scheme for the stochastic simulation of quantum spin dynamics", SciPost Physics 11 3, 048 (2021).
 Rouven Koch and Jose L. Lado, "Neural network enhanced hybrid quantum many-body dynamical distributions", Physical Review Research 3 3, 033102 (2021).
 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).
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