Neural Projected Quantum Dynamics: a systematic study

Luca Gravina; Vincenzo Savona; Filippo Vicentini

doi:10.22331/q-2025-07-22-1803

Neural Projected Quantum Dynamics: a systematic study

Luca Gravina^1,2, Vincenzo Savona^1,2, and Filippo Vicentini^3,4

¹Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
²Center for Quantum Science and Engineering, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
³CPHT, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France.
⁴Collège de France, Université PSL, 11 place Marcelin Berthelot, 75005 Paris, France

Published:	2025-07-22, volume 9, page 1803
Editor:	Alessio Benavoli
Eprint:	arXiv:2410.10720v3
Doi:	https://doi.org/10.22331/q-2025-07-22-1803
Citation:	Quantum 9, 1803 (2025).

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

Abstract

We investigate the challenge of classical simulation of unitary quantum dynamics with variational Monte Carlo approaches, addressing the instabilities and high computational demands of existing methods. By systematically analyzing the convergence of stochastic infidelity optimizations, examining the variance properties of key stochastic estimators, and evaluating the error scaling of multiple dynamical discretization schemes, we provide a thorough formalization and significant improvements to the projected time-dependent Variational Monte Carlo (p-tVMC) method. We benchmark our approach on a two-dimensional Ising quench, achieving state-of-the-art performance. This work establishes p-tVMC as a powerful framework for simulating the dynamics of large-scale two-dimensional quantum systems, surpassing alternative VMC strategies on the investigated benchmark problems.

Featured image: Illustrative representation of the generalized p-tVMC scheme. Here, the generator of the dynamics $e^{\hat{\Lambda}\text{d}t}$ is expanded as $e^{\hat{\Lambda}\text{d}t} = \hat{V}^{-1}\hat{U}$ and the variational state $|\psi_{\theta}\rangle$ is optimized to that $\hat{V} |\psi_{\theta}\rangle \mapsto \hat{U} |\psi_{\theta(t)}\rangle$. Upon a successful optimization $$
|\psi_{\theta(t+\text{d}t)}\rangle \equiv |\psi_{\theta^*}\rangle \approx \hat{V}^{-1}\hat{U}|\psi_{\theta(t)}\rangle \approx e^{\hat{\Lambda}\text{d}t}|\psi_{\theta(t)}\rangle
$$ as desired. Here, the first approximation encapsulates the optimization error, the second, the discretization error.

NeuralQXLab / ptvmc-systematic-study at GitHub

► BibTeX data

@article{Gravina2025neuralprojected,
  doi = {10.22331/q-2025-07-22-1803},
  url = {https://doi.org/10.22331/q-2025-07-22-1803},
  title = {Neural {P}rojected {Q}uantum {D}ynamics: a systematic study},
  author = {Gravina, Luca and Savona, Vincenzo and Vicentini, Filippo},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1803},
  month = jul,
  year = {2025}
}

► References

[1] I.M. Georgescu, S. Ashhab, and Franco Nori. ``Quantum simulation''. Reviews of Modern Physics 86, 153–185 (2014).
https://doi.org/10.1103/revmodphys.86.153

[2] John Preskill. ``Quantum Computing in the NISQ era and beyond''. Quantum 2, 79 (2018). url: https://doi.org/10.22331/q-2018-08-06-79.
https://doi.org/10.22331/q-2018-08-06-79

[3] Olivier Ezratty. ``Understanding quantum technologies 2023'' (2021) arXiv:arxiv:2111.15352.
arXiv:2111.15352

[4] Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C. Benjamin. ``Theory of variational quantum simulation''. Quantum 3, 191 (2019).
https://doi.org/10.22331/q-2019-10-07-191

[5] Mari Carmen Bañuls. ``Tensor network algorithms: A route map''. Annual Review of Condensed Matter Physics 14, 173–191 (2023).
https://doi.org/10.1146/annurev-conmatphys-040721-022705

[6] Román Orús. ``Tensor networks for complex quantum systems''. Nature Reviews Physics 1, 538–550 (2019).
https://doi.org/10.1038/s42254-019-0086-7

[7] U. Schollwöck. ``The density-matrix renormalization group''. Reviews of Modern Physics 77, 259–315 (2005).
https://doi.org/10.1103/revmodphys.77.259

[8] Noa Feldman, Augustine Kshetrimayum, Jens Eisert, and Moshe Goldstein. ``Entanglement estimation in tensor network states via sampling''. PRX Quantum 3 (2022).
https://doi.org/10.1103/prxquantum.3.030312

[9] J. Eisert. ``Entanglement and tensor network states'' (2013) arXiv:1308.3318.
arXiv:1308.3318

[10] A. H. Werner, D. Jaschke, P. Silvi, M. Kliesch, T. Calarco, J. Eisert, and S. Montangero. ``Positive tensor network approach for simulating open quantum many-body systems''. Phys. Rev. Lett. 116, 237201 (2016).
https://doi.org/10.1103/PhysRevLett.116.237201

[11] J. Ignacio Cirac, David Pérez-García, Norbert Schuch, and Frank Verstraete. ``Matrix product states and projected entangled pair states: Concepts, symmetries, theorems''. Reviews of Modern Physics 93 (2021).
https://doi.org/10.1103/revmodphys.93.045003

[12] Sebastian Paeckel, Thomas Köhler, Andreas Swoboda, Salvatore R. Manmana, Ulrich Schollwöck, and Claudius Hubig. ``Time-evolution methods for matrix-product states''. Annals of Physics 411, 167998 (2019).
https://doi.org/10.1016/j.aop.2019.167998

[13] 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

[14] Ulrich Schollwöck. ``The density-matrix renormalization group in the age of matrix product states''. Annals of Physics 326, 96–192 (2011).
https://doi.org/10.1016/j.aop.2010.09.012

[15] Michael Lubasch, J Ignacio Cirac, and Mari-Carmen Bañuls. ``Unifying projected entangled pair state contractions''. New Journal of Physics 16, 033014 (2014).
https://doi.org/10.1088/1367-2630/16/3/033014

[16] F. Verstraete and J. I. Cirac. ``Renormalization algorithms for quantum-many body systems in two and higher dimensions'' (2004) arXiv:cond-mat/0407066.
arXiv:cond-mat/0407066

[17] L. Tagliacozzo, G. Evenbly, and G. Vidal. ``Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law''. Phys. Rev. B 80, 235127 (2009).
https://doi.org/10.1103/PhysRevB.80.235127

[18] Timo Felser, Simone Notarnicola, and Simone Montangero. ``Efficient tensor network ansatz for high-dimensional quantum many-body problems''. Physical Review Letters 126 (2021).
https://doi.org/10.1103/physrevlett.126.170603

[19] Pietro Silvi, Ferdinand Tschirsich, Matthias Gerster, Johannes Jünemann, Daniel Jaschke, Matteo Rizzi, and Simone Montangero. ``The tensor networks anthology: Simulation techniques for many-body quantum lattice systems''. SciPost Physics Lecture Notes (2019).
https://doi.org/10.21468/scipostphyslectnotes.8

[20] Daniel Jaschke, Simone Montangero, and Lincoln D Carr. ``One-dimensional many-body entangled open quantum systems with tensor network methods''. Quantum Science and Technology 4, 013001 (2018).
https://doi.org/10.1088/2058-9565/aae724

[21] Daniel Jaschke, Michael L. Wall, and Lincoln D. Carr. ``Open source matrix product states: Opening ways to simulate entangled many-body quantum systems in one dimension''. Computer Physics Communications 225, 59–91 (2018).
https://doi.org/10.1016/j.cpc.2017.12.015

[22] G. Evenbly and G. Vidal. ``Tensor network states and geometry''. Journal of Statistical Physics 145, 891–918 (2011).
https://doi.org/10.1007/s10955-011-0237-4

[23] 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

[24] Or Sharir, Amnon Shashua, and Giuseppe Carleo. ``Neural tensor contractions and the expressive power of deep neural quantum states''. Phys. Rev. B 106, 205136 (2022).
https://doi.org/10.1103/PhysRevB.106.205136

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

[26] Julio Ureña, Antonio Sojo, Juani Bermejo-Vega, and Daniel Manzano. ``Entanglement detection with classical deep neural networks''. Scientific Reports 14 (2024).
https://doi.org/10.1038/s41598-024-68213-0

[27] Giacomo Passetti and Dante M. Kennes. ``Entanglement transition in deep neural quantum states'' (2023) arXiv:2312.11941.
arXiv:2312.11941

[28] Giacomo Torlai and Roger G. Melko. ``Latent space purification via neural density operators''. Physical Review Letters 120 (2018).
https://doi.org/10.1103/physrevlett.120.240503

[29] Filippo Vicentini, Riccardo Rossi, and Giuseppe Carleo. ``Positive-definite parametrization of mixed quantum states with deep neural networks'' (2022).
arXiv:2206.13488

[30] 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 (2022).
https://doi.org/10.1103/physrevlett.128.090501

[31] Filippo Vicentini, Alberto Biella, Nicolas Regnault, and Cristiano Ciuti. ``Variational neural-network ansatz for steady states in open quantum systems''. Physical Review Letters 122 (2019).
https://doi.org/10.1103/physrevlett.122.250503

[32] Debbie Eeltink, Filippo Vicentini, and Vincenzo Savona. ``Variational dynamics of open quantum systems in phase space'' (2023).
arXiv:2307.07429

[33] Moritz Reh, Markus Schmitt, and Martin Gärttner. ``Time-dependent variational principle for open quantum systems with artificial neural networks''. Phys. Rev. Lett. 127, 230501 (2021).
https://doi.org/10.1103/PhysRevLett.127.230501

[34] Sidhartha Dash, Luca Gravina, Filippo Vicentini, Michel Ferrero, and Antoine Georges. ``Efficiency of neural quantum states in light of the quantum geometric tensor''. Communications Physics 8 (2025).
https://doi.org/10.1038/s42005-025-02005-4

[35] Haimeng Zhao, Giuseppe Carleo, and Filippo Vicentini. ``Empirical sample complexity of neural network mixed state reconstruction''. Quantum 8, 1358 (2024).
https://doi.org/10.22331/q-2024-05-23-1358

[36] Kenny Choo, Titus Neupert, and Giuseppe Carleo. ``Two-dimensional frustrated ${J}_{1}\text{{-}}{J}_{2}$ model studied with neural network quantum states''. Phys. Rev. B 100, 125124 (2019).
https://doi.org/10.1103/PhysRevB.100.125124

[37] Or Sharir, Yoav Levine, Noam Wies, Giuseppe Carleo, and Amnon Shashua. ``Deep autoregressive models for the efficient variational simulation of many-body quantum systems''. Phys. Rev. Lett. 124, 020503 (2020).
https://doi.org/10.1103/PhysRevLett.124.020503

[38] Dian Wu, Riccardo Rossi, Filippo Vicentini, Nikita Astrakhantsev, Federico Becca, Xiaodong Cao, Juan Carrasquilla, Francesco Ferrari, Antoine Georges, Mohamed Hibat-Allah, Masatoshi Imada, Andreas M. Läuchli, Guglielmo Mazzola, Antonio Mezzacapo, Andrew Millis, Javier Robledo Moreno, Titus Neupert, Yusuke Nomura, Jannes Nys, Olivier Parcollet, Rico Pohle, Imelda Romero, Michael Schmid, J. Maxwell Silvester, Sandro Sorella, Luca F. Tocchio, Lei Wang, Steven R. White, Alexander Wietek, Qi Yang, Yiqi Yang, Shiwei Zhang, and Giuseppe Carleo. ``Variational benchmarks for quantum many-body problems''. Science 386, 296–301 (2024).
https://doi.org/10.1126/science.adg9774

[39] Luciano Loris Viteritti, Riccardo Rende, and Federico Becca. ``Transformer variational wave functions for frustrated quantum spin systems''. Phys. Rev. Lett. 130, 236401 (2023).
https://doi.org/10.1103/PhysRevLett.130.236401

[40] Xiao Liang, Wen-Yuan Liu, Pei-Ze Lin, Guang-Can Guo, Yong-Sheng Zhang, and Lixin He. ``Solving frustrated quantum many-particle models with convolutional neural networks''. Phys. Rev. B 98, 104426 (2018).
https://doi.org/10.1103/PhysRevB.98.104426

[41] James Stokes, Javier Robledo Moreno, Eftychios A. Pnevmatikakis, and Giuseppe Carleo. ``Phases of two-dimensional spinless lattice fermions with first-quantized deep neural-network quantum states''. Phys. Rev. B 102, 205122 (2020).
https://doi.org/10.1103/PhysRevB.102.205122

[42] Attila Szabó and Claudio Castelnovo. ``Neural network wave functions and the sign problem''. Phys. Rev. Res. 2, 033075 (2020).
https://doi.org/10.1103/PhysRevResearch.2.033075

[43] Kenny Choo, Antonio Mezzacapo, and Giuseppe Carleo. ``Fermionic neural-network states for ab-initio electronic structure''. Nature Communications 11 (2020).
https://doi.org/10.1038/s41467-020-15724-9

[44] Gino Cassella, Halvard Sutterud, Sam Azadi, N. D. Drummond, David Pfau, James S. Spencer, and W. M. C. Foulkes. ``Discovering quantum phase transitions with fermionic neural networks''. Phys. Rev. Lett. 130, 036401 (2023).
https://doi.org/10.1103/PhysRevLett.130.036401

[45] Javier Robledo Moreno, Giuseppe Carleo, Antoine Georges, and James Stokes. ``Fermionic wave functions from neural-network constrained hidden states''. Proceedings of the National Academy of Sciences 119 (2022).
https://doi.org/10.1073/pnas.2122059119

[46] Giuseppe Carleo, Lorenzo Cevolani, Laurent Sanchez-Palencia, and Markus Holzmann. ``Unitary dynamics of strongly interacting bose gases with the time-dependent variational monte carlo method in continuous space''. Physical Review X 7 (2017).
https://doi.org/10.1103/physrevx.7.031026

[47] Alessandro Sinibaldi, Clemens Giuliani, Giuseppe Carleo, and Filippo Vicentini. ``Unbiasing time-dependent variational monte carlo by projected quantum evolution''. Quantum 7, 1131 (2023).
https://doi.org/10.22331/q-2023-10-10-1131

[48] James Stokes, Brian Chen, and Shravan Veerapaneni. ``Numerical and geometrical aspects of flow-based variational quantum monte carlo''. Machine Learning: Science and Technology 4, 021001 (2023).
https://doi.org/10.1088/2632-2153/acc8b9

[49] Markus Schmitt and Markus Heyl. ``Quantum many-body dynamics in two dimensions with artificial neural networks''. Physical Review Letters 125 (2020).
https://doi.org/10.1103/physrevlett.125.100503

[50] Tiago Mendes-Santos, Markus Schmitt, and Markus Heyl. ``Highly resolved spectral functions of two-dimensional systems with neural quantum states''. Phys. Rev. Lett. 131, 046501 (2023).
https://doi.org/10.1103/PhysRevLett.131.046501

[51] Ashish Joshi, Robert Peters, and Thore Posske. ``Quantum skyrmion dynamics studied by neural network quantum states''. Physical Review B 110 (2024).
https://doi.org/10.1103/physrevb.110.104411

[52] 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

[53] Linda Mauron, Zakari Denis, Jannes Nys, and Giuseppe Carleo. ``Predicting topological entanglement entropy in a rydberg analog simulator'' (2024) arXiv:2406.19872.
arXiv:2406.19872

[54] Jannes Nys, Gabriel Pescia, Alessandro Sinibaldi, and Giuseppe Carleo. ``Ab-initio variational wave functions for the time-dependent many-electron schrödinger equation''. Nature Communications 15 (2024).
https://doi.org/10.1038/s41467-024-53672-w

[55] Dennis Wagner, Andreas Klümper, and Jesko Sirker. ``Thermodynamics based on neural networks''. Physical Review B 109 (2024).
https://doi.org/10.1103/physrevb.109.155128

[56] Jannes Nys, Zakari Denis, and Giuseppe Carleo. ``Real-time quantum dynamics of thermal states with neural thermofields''. Phys. Rev. B 109, 235120 (2024).
https://doi.org/10.1103/PhysRevB.109.235120

[57] Filippo Vicentini, Riccardo Rossi, and Giuseppe Carleo. ``Positive-definite parametrization of mixed quantum states with deep neural networks'' (2022) arXiv:2206.13488.
arXiv:2206.13488

[58] Joshua Lin, Di Luo, Xiaojun Yao, and Phiala E. Shanahan. ``Real-time dynamics of the schwinger model as an open quantum system with neural density operators''. Journal of High Energy Physics 2024 (2024).
https://doi.org/10.1007/jhep06(2024)211

[59] L. F. Shampine and C. W. Gear. ``A user’s view of solving stiff ordinary differential equations''. SIAM Review 21, 1–17 (1979).
https://doi.org/10.1137/1021001

[60] Kaelan Donatella, Zakari Denis, Alexandre Le Boité, and Cristiano Ciuti. ``Dynamics with autoregressive neural quantum states: Application to critical quench dynamics''. Phys. Rev. A 108, 022210 (2023).
https://doi.org/10.1103/PhysRevA.108.022210

[61] Wenxuan Zhang, Bo Xing, Xiansong Xu, and Dario Poletti. ``Paths towards time evolution with larger neural-network quantum states'' (2024) arXiv:2406.03381.
arXiv:2406.03381

[62] Matija Medvidović and Giuseppe Carleo. ``Classical variational simulation of the quantum approximate optimization algorithm''. npj Quantum Information 7 (2021).
https://doi.org/10.1038/s41534-021-00440-z

[63] Bjarni Jónsson, Bela Bauer, and Giuseppe Carleo. ``Neural-network states for the classical simulation of quantum computing'' (2018).
arXiv:1808.05232

[64] Irene López Gutiérrez and Christian B. Mendl. ``Real time evolution with neural-network quantum states''. Quantum 6, 627 (2022).
https://doi.org/10.22331/q-2022-01-20-627

[65] Matija Medvidović and Javier Robledo Moreno. ``Neural-network quantum states for many-body physics''. The European Physical Journal Plus 139 (2024).
https://doi.org/10.1140/epjp/s13360-024-05311-y

[66] Hannah Lange, Anka Van de Walle, Atiye Abedinnia, and Annabelle Bohrdt. ``From architectures to applications: A review of neural quantum states'' (2024) arXiv:2402.09402.
arXiv:2402.09402

[67] Naomichi Hatano and Masuo Suzuki. ``Finding exponential product formulas of higher orders''. Page 37–68. Springer Berlin Heidelberg. (2005).
https://doi.org/10.1007/11526216_2

[68] Alexander Müller-Hermes, J Ignacio Cirac, and Mari Carmen Bañuls. ``Tensor network techniques for the computation of dynamical observables in one-dimensional quantum spin systems''. New Journal of Physics 14, 075003 (2012).
https://doi.org/10.1088/1367-2630/14/7/075003

[69] Cleve Moler and Charles Van Loan. ``Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later''. SIAM Review 45, 3–49 (2003).
https://doi.org/10.1137/s00361445024180

[70] Vojtech Havlicek. ``Amplitude ratios and neural network quantum states''. Quantum 7, 938 (2023).
https://doi.org/10.22331/q-2023-03-02-938

[71] Eimantas Ledinauskas and Egidijus Anisimovas. ``Scalable imaginary time evolution with neural network quantum states''. SciPost Physics 15 (2023).
https://doi.org/10.21468/scipostphys.15.6.229

[72] Diederik P. Kingma and Jimmy Ba. ``Adam: A method for stochastic optimization'' (2014) arXiv:1412.6980.
arXiv:1412.6980

[73] Reuven Y. Rubinstein and Dirk P. Kroese. ``Simulation and the monte carlo method''. Chapter 5. Wiley. (2016).
https://doi.org/10.1002/9781118631980

[74] Rajesh Ranganath, Sean Gerrish, and David Blei. ``Black box variational inference''. In Artificial intelligence and statistics. Pages 814–822. PMLR (2014).
https://doi.org/10.48550/arXiv.1401.0118

[75] James Martens and Ilya Sutskever. ``Training deep and recurrent networks with hessian-free optimization''. Page 479–535. Springer Berlin Heidelberg. (2012).
https://doi.org/10.1007/978-3-642-35289-8_27

[76] James Martens. ``New insights and perspectives on the natural gradient method''. Journal of Machine Learning Research 21, 1–76 (2020). url: http://jmlr.org/papers/v21/17-678.html.
http://jmlr.org/papers/v21/17-678.html

[77] Jorge Nocedal and Stephen J. Wright. ``Numerical optimization''. Volume 2 of Springer Series in Operations Research and Financial Engineering. Springer New York. (2006).
https://doi.org/10.1007/978-0-387-40065-5

[78] Ran Cheng. ``Quantum geometric tensor (fubini-study metric) in simple quantum system: A pedagogical introduction'' (2010) arXiv:1012.1337.
arXiv:1012.1337

[79] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo. ``Quantum natural gradient''. Quantum 4, 269 (2020).
https://doi.org/10.22331/q-2020-05-25-269

[80] Tom Heskes. ``On “natural” learning and pruning in multilayered perceptrons''. Neural Computation 12, 881–901 (2000).
https://doi.org/10.1162/089976600300015637

[81] Roger Grosse and Ruslan Salakhudinov. ``Scaling up natural gradient by sparsely factorizing the inverse fisher matrix''. In Francis Bach and David Blei, editors, Proceedings of the 32nd International Conference on Machine Learning. Volume 37 of Proceedings of Machine Learning Research, pages 2304–2313. Lille, France (2015). PMLR. url: https://proceedings.mlr.press/v37/grosse15.html.
https://proceedings.mlr.press/v37/grosse15.html

[82] James Martens and Roger Grosse. ``Optimizing neural networks with kronecker-factored approximate curvature''. In Francis Bach and David Blei, editors, Proceedings of the 32nd International Conference on Machine Learning. Volume 37 of Proceedings of Machine Learning Research, pages 2408–2417. Lille, France (2015). PMLR. url: https://proceedings.mlr.press/v37/martens15.html.
https://proceedings.mlr.press/v37/martens15.html

[83] Roger Grosse and James Martens. ``A kronecker-factored approximate fisher matrix for convolution layers''. In Maria Florina Balcan and Kilian Q. Weinberger, editors, Proceedings of The 33rd International Conference on Machine Learning. Volume 48 of Proceedings of Machine Learning Research, pages 573–582. New York, New York, USA (2016). PMLR. url: https://proceedings.mlr.press/v48/grosse16.html.
https://proceedings.mlr.press/v48/grosse16.html

[84] Y. Ollivier. ``Riemannian metrics for neural networks i: feedforward networks''. Information and Inference 4, 108–153 (2015).
https://doi.org/10.1093/imaiai/iav006

[85] Shun-ichi Amari, Ryo Karakida, and Masafumi Oizumi. ``Fisher information and natural gradient learning in random deep networks''. In Kamalika Chaudhuri and Masashi Sugiyama, editors, Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics. Volume 89 of Proceedings of Machine Learning Research, pages 694–702. PMLR (2019). url: https://proceedings.mlr.press/v89/amari19a.html.
https://proceedings.mlr.press/v89/amari19a.html

[86] Ryo Karakida and Kazuki Osawa. ``Understanding approximate fisher information for fast convergence of natural gradient descent in wide neural networks*''. Journal of Statistical Mechanics: Theory and Experiment 2021, 124010 (2021).
https://doi.org/10.1088/1742-5468/ac3ae3

[87] Qinxun Bai, Steven Rosenberg, and Wei Xu. ``A geometric understanding of natural gradient'' (2022) arXiv:2202.06232.
arXiv:2202.06232

[88] Ao Chen and Markus Heyl. ``Empowering deep neural quantum states through efficient optimization''. Nature Physics 20, 1476–1481 (2024).
https://doi.org/10.1038/s41567-024-02566-1

[89] K. B. Petersen and M. S. Pedersen. ``The matrix cookbook'' (2012).

[90] Riccardo Rende, Luciano Loris Viteritti, Lorenzo Bardone, Federico Becca, and Sebastian Goldt. ``A simple linear algebra identity to optimize large-scale neural network quantum states''. Communications Physics 7 (2024).
https://doi.org/10.1038/s42005-024-01732-4

[91] Arthur Jacot, Franck Gabriel, and Clément Hongler. ``Neural tangent kernel: convergence and generalization in neural networks''. In Proceedings of the 32nd International Conference on Neural Information Processing Systems. Page 8580–8589. NIPS'18Red Hook, NY, USA (2018). Curran Associates Inc.
https://doi.org/10.48550/arXiv.1806.07572

[92] Roman Novak, Jascha Sohl-Dickstein, and Samuel S Schoenholz. ``Fast finite width neural tangent kernel''. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari, Gang Niu, and Sivan Sabato, editors, Proceedings of the 39th International Conference on Machine Learning. Volume 162 of Proceedings of Machine Learning Research, pages 17018–17044. PMLR (2022). url: https://proceedings.mlr.press/v162/novak22a.html.
https://proceedings.mlr.press/v162/novak22a.html

[93] Julien Gacon, Jannes Nys, Riccardo Rossi, Stefan Woerner, and Giuseppe Carleo. ``Variational quantum time evolution without the quantum geometric tensor''. Phys. Rev. Res. 6, 013143 (2024).
https://doi.org/10.1103/PhysRevResearch.6.013143

[94] James Martens. ``Deep learning via hessian-free optimization''. In Proceedings of the 27th International Conference on International Conference on Machine Learning. Page 735–742. ICML'10Madison, WI, USA (2010). Omnipress. url: https://dl.acm.org/doi/10.5555/3104322.3104416.
https://dl.acm.org/doi/10.5555/3104322.3104416

[95] Gerhard Wanner and Ernst Hairer. ``Solving ordinary differential equations ii''. Volume 375. Springer Berlin Heidelberg New York. (1996).
https://doi.org/10.1007/978-3-642-05221-7

[96] Luciano Loris Viteritti, Riccardo Rende, Alberto Parola, Sebastian Goldt, and Federico Becca. ``Transformer wave function for the shastry-sutherland model: emergence of a spin-liquid phase'' (2023) arXiv:2311.16889.
https://doi.org/10.1103/PhysRevB.111.134411
arXiv:2311.16889

[97] Piotr Czarnik, Jacek Dziarmaga, and Philippe Corboz. ``Time evolution of an infinite projected entangled pair state: An efficient algorithm''. Phys. Rev. B 99, 035115 (2019).
https://doi.org/10.1103/PhysRevB.99.035115

[98] 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 Phys. CodebasesPage 7 (2022).
https://doi.org/10.21468/SciPostPhysCodeb.7

[99] Giuseppe Carleo, Kenny Choo, Damian Hofmann, James E. T. Smith, Tom Westerhout, Fabien Alet, Emily J. Davis, Stavros Efthymiou, Ivan Glasser, Sheng-Hsuan Lin, Marta Mauri, Guglielmo Mazzola, Christian B. Mendl, Evert van Nieuwenburg, Ossian O'Reilly, Hugo Théveniaut, Giacomo Torlai, Filippo Vicentini, and Alexander Wietek. ``Netket: A machine learning toolkit for many-body quantum systems''. SoftwareXPage 100311 (2019).
https://doi.org/10.1016/j.softx.2019.100311

[100] Dion Häfner and Filippo Vicentini. ``mpi4jax: Zero-copy mpi communication of jax arrays''. Journal of Open Source Software 6, 3419 (2021).
https://doi.org/10.21105/joss.03419

[101] James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman-Milne, and Qiao Zhang. ``JAX: composable transformations of Python+NumPy programs''. https://github.com/google/jax (2018). Version 0.3.13.
https://github.com/google/jax

[102] Patrick Kidger and Cristian Garcia. ``Equinox: neural networks in jax via callable pytrees and filtered transformations'' (2021) arXiv:2111.00254.
arXiv:2111.00254

[103] Jonathan Heek, Anselm Levskaya, Avital Oliver, Marvin Ritter, Bertrand Rondepierre, Andreas Steiner, and Marc van Zee. ``Flax: A neural network library and ecosystem for JAX''. https://github.com/google/flax (2024). Version 0.10.6.
https://github.com/google/flax

[104] J.R. Johansson, P.D. Nation, and F. Nori. ``QuTiP: An open-source Python framework for the dynamics of open quantum systems''. Computer Physics Communications 183, 1760–1772 (2012).
https://doi.org/10.1016/j.cpc.2012.02.021

[105] J.R. Johansson, P.D. Nation, and F. Nori. ``QuTiP 2: A Python framework for the dynamics of open quantum systems''. Computer Physics Communications 184, 1234–1240 (2013).
https://doi.org/10.1016/j.cpc.2012.11.019

[106] Richard P. Stanley. ``Enumerative combinatorics, volume 2''. Volume 2 of Cambridge Studies in Advanced Mathematics. Cambridge University Press. (1999).
https://doi.org/10.1017/CBO9780511609589

[107] Uri M. Ascher, Steven J. Ruuth, and Brian T. R. Wetton. ``Implicit-explicit methods for time-dependent partial differential equations''. SIAM Journal on Numerical Analysis 32, 797–823 (1995).
https://doi.org/10.1137/0732037

[108] Uri M. Ascher, Steven J. Ruuth, and Raymond J. Spiteri. ``Implicit-explicit runge-kutta methods for time-dependent partial differential equations''. Applied Numerical Mathematics 25, 151–167 (1997).
https://doi.org/10.1016/s0168-9274(97)00056-1

[109] Ruichen Li, Haotian Ye, Du Jiang, Xuelan Wen, Chuwei Wang, Zhe Li, Xiang Li, Di He, Ji Chen, Weiluo Ren, and Liwei Wang. ``A computational framework for neural network-based variational monte carlo with forward laplacian''. Nature Machine Intelligence (2024).
https://doi.org/10.1038/s42256-024-00794-x

[110] Fabrizio Minganti, Adam Miranowicz, Ravindra W. Chhajlany, and Franco Nori. ``Quantum exceptional points of non-hermitian hamiltonians and liouvillians: The effects of quantum jumps''. Physical Review A 100 (2019).
https://doi.org/10.1103/physreva.100.062131

Cited by

[1] Thomas Spriggs, Eliska Greplova, Juan Carrasquilla, and Jannes Nys, "Accurate Ground States of SU(2) Lattice Gauge Theory in 2+1D and 3+1D", Physical Review Letters 136 10, 101902 (2026).

[2] Andrew Jreissaty, Hang Zhang, Jairo C. Quijano, Juan Carrasquilla, and Roeland Wiersema, "Entanglement and optimization within autoregressive neural quantum states", Physical Review Research 8 1, 013147 (2026).

[3] Antoine Misery, Luca Gravina, Alessandro Santini, and Filippo Vicentini, "Looking elsewhere: improving variational Monte Carlo gradients by importance sampling", Machine Learning: Science and Technology 7 1, 015035 (2026).

[4] Adrien Kahn, Luca Gravina, and Filippo Vicentini, "Variational subspace methods and application to improving variational Monte Carlo dynamics", Quantum 10, 2082 (2026).

[5] Linda Mauron, Zakari Denis, Jannes Nys, and Giuseppe Carleo, "Predicting topological entanglement entropy in a Rydberg analogue simulator", Nature Physics 21 8, 1332 (2025).

[6] Louis Sharma, Ahmedeo Shokry, Rajah Nutakki, Olivier Simard, Michel Ferrero, and Filippo Vicentini, "Comparing symmetrized determinant neural quantum states for the Hubbard model", Physical Review B 113 24, 245104 (2026).

[7] Raffaele Salioni, Rocco Martinazzo, Davide Emilio Galli, and Christian Apostoli, "Adaptive quantum dynamics with the time-dependent variational Monte Carlo method", Physical Review B 113 1, 014408 (2026).

[8] Ao Chen, Vighnesh Dattatraya Naik, and Markus Heyl, "Convolutional transformer wave functions", Physical Review Research 8 2, L022040 (2026).

[9] Elad Parnes, Nir Barnea, Giuseppe Carleo, Alessandro Lovato, Noemi Rocco, and Xilin Zhang, "Nuclear Responses with Neural-Network Quantum States", Physical Review Letters 136 3, 032501 (2026).

[10] Jia-Qi Wang, Rong-Qiang He, and Zhong-Yi Lu, "Generalized Lanczos method for systematic optimization of neural-network quantum states", Physical Review B 113 8, 085120 (2026).

[11] Alessandro Sinibaldi, Antonio Francesco Mello, Mario Collura, and Giuseppe Carleo, "Nonstabilizerness of neural quantum states", Physical Review Research 7 4, 043289 (2025).

[12] Bizi Huang, Weizhong Fu, and Ji Chen, "Stochastic representation of time-evolving neural network-based wavefunctions", The Journal of Chemical Physics 163 24, 244107 (2025).

[13] Shaojun Gui, Tak-San Ho, and Herschel Rabitz, "Control simulations of many-body quantum systems by a synergism of discrete real-time learning and optimal control theory", The Journal of Chemical Physics 163 10, 104108 (2025).

[14] Rajah P. Nutakki, Ahmedeo Shokry, and Filippo Vicentini, "Design principles of deep translationally symmetric neural quantum states for frustrated magnets", Physical Review Research 7 4, 043099 (2025).

[15] Alessandro Sinibaldi, Douglas Hendry, Filippo Vicentini, and Giuseppe Carleo, "Time-Dependent Neural Galerkin Method for Quantum Dynamics", Physical Review Letters 136 12, 120402 (2026).

[16] Anka Van de Walle, Markus Schmitt, and Annabelle Bohrdt, "Many-body dynamics with explicitly time-dependent neural quantum states", Machine Learning: Science and Technology 6 4, 045011 (2025).

[17] Tarun Advaith Kumar, Leon Balents, Timothy H. Hsieh, and Roger G. Melko, "Autoregressive typical thermal states", Annals of Physics 487, 170362 (2026).

[18] Ilya Shirokov, Viacheslav Khrushchev, Filipp Uskov, Ivan Dudinets, Igor Ermakov, and Oleg Lychkovskiy, "Recursion method for quench dynamics: strengths and limitations", arXiv:2503.24362, (2025).

[19] Sidhartha Dash, Luca Gravina, Filippo Vicentini, Michel Ferrero, and Antoine Georges, "Efficiency of neural quantum states in light of the quantum geometric tensor", Communications Physics 8 1, 92 (2025).

[20] Douglas Hendry, Alessandro Sinibaldi, and Giuseppe Carleo, "Grassmann Variational Monte Carlo with neural wave functions", arXiv:2507.10287, (2025).

[21] Dingzu Wang, Wenxuan Zhang, Xiansong Xu, and Dario Poletti, "Continuous-time parametrization of neural quantum states for quantum dynamics", arXiv:2507.08418, (2025).

[22] DinhDuy Vu, Dominik S. Kufel, Jack Kemp, Lode Pollet, Chris R. Laumann, and Norman Y. Yao, "Optimizing the dynamical preparation of quantum spin lakes on the ruby lattice", arXiv:2512.09040, (2025).

[23] Hrvoje Vrcan and Johan H. Mentink, "Instability of explicit time integration for strongly quenched dynamics with neural quantum states", arXiv:2507.17421, (2025).

[24] Ahmedeo Shokry, Alessandro Santini, and Filippo Vicentini, "When Less is More: Approximating the Quantum Geometric Tensor with Block Structures", arXiv:2510.08430, (2025).

[25] Jia-Lin Chen, Zhen Fan, Canhui Yan, Yantao Wu, and Tao Xiang, "Resolving support-mismatch by local basis rotation in variational Monte Carlo", arXiv:2606.23657, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-12 02:39:59) and SAO/NASA ADS (last updated successfully 2026-08-12 02:40:00). 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.