Unbiasing time-dependent Variational Monte Carlo by projected quantum evolution
1Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
2Center for Quantum Science and Engineering, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
3CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
| Published: | 2023-10-10, volume 7, page 1131 |
| Eprint: | arXiv:2305.14294v3 |
| Doi: | https://doi.org/10.22331/q-2023-10-10-1131 |
| Citation: | Quantum 7, 1131 (2023). |
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
We analyze the accuracy and sample complexity of variational Monte Carlo approaches to simulate the dynamics of many-body quantum systems classically. By systematically studying the relevant stochastic estimators, we are able to: (i) prove that the most used scheme, the time-dependent Variational Monte Carlo (tVMC), is affected by a systematic statistical bias or exponential sample complexity when the wave function contains some (possibly approximate) zeros, an important case for fermionic systems and quantum information protocols; (ii) show that a different scheme based on the solution of an optimization problem at each time step is free from such problems; (iii) improve the sample complexity of this latter approach by several orders of magnitude with respect to previous proofs of concept. Finally, we apply our advancements to study the high-entanglement phase in a protocol of non-Clifford unitary dynamics with local random measurements in 2D, first benchmarking on small spin lattices and then extending to large systems.
Featured image: Sketch of the failure of the tVMC when the state features zeros (red) and of the dynamics generated by the p-tVMC algorithm (blue). In p-tVMC, the optimization problem to solve consists in minimizing a distance in the Hilbert space, which is the infidelity $\mathcal{I}$ (as shown in the right panel).
Popular summary
In this article, we first show that the classical implementation of TDVP, known as time-dependent Variational Monte Carlo (tVMC), can be affected by a statistical bias or an exponential sampling complexity when the wave function contains possibly approximate nodes. Second, we derive an extension of the implicit scheme that lowers its computational cost by several orders of magnitude and we name it projected tVMC (p-tVMC). In the final part, we apply the p-tVMC to investigate a system evolving via non-Clifford unitary dynamics with local random measurements in 2D, which is a paradigmatic model for entanglement phase transition that cannot be fully studied with other state-of-the-art techniques.
► BibTeX data
► References
[1] I. M. Georgescu, S. Ashhab, and F. Nori. ``Quantum simulation''. Rev. Mod. Phys. 86, 153–185 (2014). doi: 10.1103/RevModPhys.86.153.
https://doi.org/10.1103/RevModPhys.86.153
[2] F. Minganti, I. I. Arkhipov, A. Miranowicz, and F. Nori. ``Continuous dissipative phase transitions with or without symmetry breaking''. New Journal of Physics 23, 122001 (2021). doi: 10.1088/1367-2630/ac3db8.
https://doi.org/10.1088/1367-2630/ac3db8
[3] M. A. Nielsen and I. L. Chuang. ``Quantum computation and quantum information: 10th anniversary edition''. Cambridge University Press. (2010). doi: 10.1017/CBO9780511976667.
https://doi.org/10.1017/CBO9780511976667
[4] B. Skinner, J. Ruhman, and A. Nahum. ``Measurement-induced phase transitions in the dynamics of entanglement''. Phys. Rev. X 9, 031009 (2019). doi: 10.1103/PhysRevX.9.031009.
https://doi.org/10.1103/PhysRevX.9.031009
[5] Y. Li, X. Chen, and M. P. A. Fisher. ``Measurement-driven entanglement transition in hybrid quantum circuits''. Phys. Rev. B 100, 134306 (2019). doi: 10.1103/PhysRevB.100.134306.
https://doi.org/10.1103/PhysRevB.100.134306
[6] Q. Tang and W. Zhu. ``Measurement-induced phase transition: A case study in the nonintegrable model by density-matrix renormalization group calculations''. Phys. Rev. Res. 2, 013022 (2020). doi: 10.1103/PhysRevResearch.2.013022.
https://doi.org/10.1103/PhysRevResearch.2.013022
[7] X. Turkeshi, R. Fazio, and M. Dalmonte. ``Measurement-induced criticality in $(2+1)$-dimensional hybrid quantum circuits''. Phys. Rev. B 102, 014315 (2020). doi: 10.1103/PhysRevB.102.014315.
https://doi.org/10.1103/PhysRevB.102.014315
[8] X. Turkeshi, A. Biella, R. Fazio, M. Dalmonte, and M. Schiró. ``Measurement-induced entanglement transitions in the quantum ising chain: From infinite to zero clicks''. Phys. Rev. B 103, 224210 (2021). doi: 10.1103/PhysRevB.103.224210.
https://doi.org/10.1103/PhysRevB.103.224210
[9] A. Lavasani, Y. Alavirad, and M. Barkeshli. ``Topological order and criticality in $(2+1)\mathrm{D}$ monitored random quantum circuits''. Phys. Rev. Lett. 127, 235701 (2021). doi: 10.1103/PhysRevLett.127.235701.
https://doi.org/10.1103/PhysRevLett.127.235701
[10] O. Lunt, M. Szyniszewski, and A. Pal. ``Measurement-induced criticality and entanglement clusters: A study of one-dimensional and two-dimensional clifford circuits''. Phys. Rev. B 104, 155111 (2021). doi: 10.1103/PhysRevB.104.155111.
https://doi.org/10.1103/PhysRevB.104.155111
[11] H. Liu, T. Zhou, and X. Chen. ``Measurement-induced entanglement transition in a two-dimensional shallow circuit''. Phys. Rev. B 106, 144311 (2022). doi: 10.1103/PhysRevB.106.144311.
https://doi.org/10.1103/PhysRevB.106.144311
[12] X. Turkeshi, M. Dalmonte, R. Fazio, and M. Schiró. ``Entanglement transitions from stochastic resetting of non-hermitian quasiparticles''. Phys. Rev. B 105, L241114 (2022). doi: 10.1103/PhysRevB.105.L241114.
https://doi.org/10.1103/PhysRevB.105.L241114
[13] P. Sierant, M. Schiró, M. Lewenstein, and X. Turkeshi. ``Measurement-induced phase transitions in $(d+1)$-dimensional stabilizer circuits''. Phys. Rev. B 106, 214316 (2022). doi: 10.1103/PhysRevB.106.214316.
https://doi.org/10.1103/PhysRevB.106.214316
[14] J. C. Hoke et al. ``Quantum information phases in space-time: measurement-induced entanglement and teleportation on a noisy quantum processor'' (2023). arXiv:2303.04792.
arXiv:2303.04792
[15] S. R. White. ``Density matrix formulation for quantum renormalization groups''. Phys. Rev. Lett. 69, 2863–2866 (1992). doi: 10.1103/PhysRevLett.69.2863.
https://doi.org/10.1103/PhysRevLett.69.2863
[16] R. Orús. ``A practical introduction to tensor networks: Matrix product states and projected entangled pair states''. Annals of physics 349, 117–158 (2014). doi: 10.1016/j.aop.2014.06.013.
https://doi.org/10.1016/j.aop.2014.06.013
[17] D. Ceperley and B. Alder. ``Quantum monte carlo''. Science 231, 555–560 (1986). doi: 10.1126/science.231.4738.555.
https://doi.org/10.1126/science.231.4738.555
[18] W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal. ``Quantum monte carlo simulations of solids''. Rev. Mod. Phys. 73, 33–83 (2001). doi: 10.1103/RevModPhys.73.33.
https://doi.org/10.1103/RevModPhys.73.33
[19] M. Troyer and U.-J. Wiese. ``Computational complexity and fundamental limitations to fermionic quantum monte carlo simulations''. Phys. Rev. Lett. 94, 170201 (2005). doi: 10.1103/PhysRevLett.94.170201.
https://doi.org/10.1103/PhysRevLett.94.170201
[20] K. Choo, A. Mezzacapo, and G. Carleo. ``Fermionic neural-network states for ab-initio electronic structure''. Nature Communications 11 (2020). doi: 10.1038/s41467-020-15724-9.
https://doi.org/10.1038/s41467-020-15724-9
[21] D. Pfau, J. S. Spencer, A. G. D. G. Matthews, and W. M. C. Foulkes. ``Ab initio solution of the many-electron schrödinger equation with deep neural networks''. Phys. Rev. Research 2, 033429 (2020). doi: 10.1103/PhysRevResearch.2.033429.
https://doi.org/10.1103/PhysRevResearch.2.033429
[22] 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). doi: 10.1103/PhysRevB.96.205152.
https://doi.org/10.1103/PhysRevB.96.205152
[23] J. Stokes, J. R. Moreno, E. A. Pnevmatikakis, and G. Carleo. ``Phases of two-dimensional spinless lattice fermions with first-quantized deep neural-network quantum states''. Phys. Rev. B 102, 205122 (2020). doi: 10.1103/PhysRevB.102.205122.
https://doi.org/10.1103/PhysRevB.102.205122
[24] J. Nys and G. Carleo. ``Variational solutions to fermion-to-qubit mappings in two spatial dimensions''. Quantum 6, 833 (2022). doi: 10.22331/q-2022-10-13-833.
https://doi.org/10.22331/q-2022-10-13-833
[25] K. Choo, T. Neupert, and G. Carleo. ``Two-dimensional frustrated ${J}_{1}\text{{-}}{J}_{2}$ model studied with neural network quantum states''. Phys. Rev. B 100, 125124 (2019). doi: 10.1103/PhysRevB.100.125124.
https://doi.org/10.1103/PhysRevB.100.125124
[26] O. Sharir, Y. Levine, N. Wies, G. Carleo, and A. Shashua. ``Deep autoregressive models for the efficient variational simulation of many-body quantum systems''. Phys. Rev. Lett. 124, 020503 (2020). doi: 10.1103/PhysRevLett.124.020503.
https://doi.org/10.1103/PhysRevLett.124.020503
[27] D. Wu et al. ``Variational benchmarks for quantum many-body problems'' (2023) arXiv:2302.04919.
arXiv:2302.04919
[28] W. L. McMillan. ``Ground state of liquid ${\mathrm{He}}^{4}$''. Phys. Rev. 138, A442–A451 (1965). doi: 10.1103/PhysRev.138.A442.
https://doi.org/10.1103/PhysRev.138.A442
[29] X. Yuan, S. Endo, Q. Zhao, Y. Li, and S. C. Benjamin. ``Theory of variational quantum simulation''. Quantum 3, 191 (2019). doi: 10.22331/q-2019-10-07-191.
https://doi.org/10.22331/q-2019-10-07-191
[30] G. Carleo, F. Becca, M. Schiró, and M. Fabrizio. ``Localization and glassy dynamics of many-body quantum systems''. Scientific Reports 2 (2012). doi: 10.1038/srep00243.
https://doi.org/10.1038/srep00243
[31] G. Carleo, L. Cevolani, L. Sanchez-Palencia, and M. Holzmann. ``Unitary dynamics of strongly interacting bose gases with the time-dependent variational monte carlo method in continuous space''. Phys. Rev. X 7, 031026 (2017). doi: 10.1103/PhysRevX.7.031026.
https://doi.org/10.1103/PhysRevX.7.031026
[32] B. Jónsson, B. Bauer, and G. Carleo. ``Neural-network states for the classical simulation of quantum computing'' (2018). arXiv:1808.05232.
arXiv:1808.05232
[33] M. Medvidović and G. Carleo. ``Classical variational simulation of the quantum approximate optimization algorithm''. npj Quantum Information 7, 1–7 (2021). doi: 10.1038/s41534-021-00440-z.
https://doi.org/10.1038/s41534-021-00440-z
[34] S. Barison, F. Vicentini, and G. Carleo. ``An efficient quantum algorithm for the time evolution of parameterized circuits''. Quantum 5, 512 (2021). doi: 10.22331/q-2021-07-28-512.
https://doi.org/10.22331/q-2021-07-28-512
[35] K. Donatella, Z. Denis, A. Le Boité, and C. Ciuti. ``Dynamics with autoregressive neural quantum states: Application to critical quench dynamics''. Phys. Rev. A 108, 022210 (2023). doi: 10.1103/PhysRevA.108.022210.
https://doi.org/10.1103/PhysRevA.108.022210
[36] I. L. Gutiérrez and C. B. Mendl. ``Real time evolution with neural-network quantum states''. Quantum 6, 627 (2022). doi: 10.22331/q-2022-01-20-627.
https://doi.org/10.22331/q-2022-01-20-627
[37] M. Schmitt and M. Heyl. ``Quantum many-body dynamics in two dimensions with artificial neural networks''. Phys. Rev. Lett. 125, 100503 (2020). doi: 10.1103/PhysRevLett.125.100503.
https://doi.org/10.1103/PhysRevLett.125.100503
[38] R. Verdel, M. Schmitt, Y.-P. Huang, P. Karpov, and M. Heyl. ``Variational classical networks for dynamics in interacting quantum matter''. Phys. Rev. B 103, 165103 (2021). doi: 10.1103/PhysRevB.103.165103.
https://doi.org/10.1103/PhysRevB.103.165103
[39] M. Schmitt, M. M. Rams, J. Dziarmaga, M. Heyl, and W. H. Zurek. ``Quantum phase transition dynamics in the two-dimensional transverse-field ising model''. Science Advances 8, eabl6850 (2022). doi: 10.1126/sciadv.abl6850.
https://doi.org/10.1126/sciadv.abl6850
[40] M. V. den Nest. ``Simulating quantum computers with probabilistic methods'' (2010). arXiv:0911.1624.
arXiv:0911.1624
[41] N. Moiseyev. ``Non-hermitian quantum mechanics''. Cambridge University Press. (2011). doi: 10.1017/CBO9780511976186.
https://doi.org/10.1017/CBO9780511976186
[42] S. Haroche and J.-M. Raimond. ``Exploring the Quantum: Atoms, Cavities, and Photons''. Oxford University Press. (2006). doi: 10.1093/acprof:oso/9780198509141.001.0001.
https://doi.org/10.1093/acprof:oso/9780198509141.001.0001
[43] A. McLachlan. ``A variational solution of the time-dependent schrodinger equation''. Molecular Physics 8, 39–44 (1964). doi: 10.1080/00268976400100041.
https://doi.org/10.1080/00268976400100041
[44] J. Stokes, J. Izaac, N. Killoran, and G. Carleo. ``Quantum Natural Gradient''. Quantum 4, 269 (2020). doi: 10.22331/q-2020-05-25-269.
https://doi.org/10.22331/q-2020-05-25-269
[45] C.-Y. Park and M. J. Kastoryano. ``Geometry of learning neural quantum states''. Phys. Rev. Res. 2, 023232 (2020). doi: 10.1103/PhysRevResearch.2.023232.
https://doi.org/10.1103/PhysRevResearch.2.023232
[46] L. Hackl, T. Guaita, T. Shi, J. Haegeman, E. Demler, and J. I. Cirac. ``Geometry of variational methods: dynamics of closed quantum systems''. SciPost Phys. 9, 048 (2020). doi: 10.21468/SciPostPhys.9.4.048.
https://doi.org/10.21468/SciPostPhys.9.4.048
[47] G. Carleo and M. Troyer. ``Solving the quantum many-body problem with artificial neural networks''. Science 355, 602–606 (2017). doi: 10.1126/science.aag2302.
https://doi.org/10.1126/science.aag2302
[48] D. Luo and B. K. Clark. ``Backflow transformations via neural networks for quantum many-body wave functions''. Phys. Rev. Lett. 122, 226401 (2019). doi: 10.1103/PhysRevLett.122.226401.
https://doi.org/10.1103/PhysRevLett.122.226401
[49] K. Mølmer, Y. Castin, and J. Dalibard. ``Monte carlo wave-function method in quantum optics''. J. Opt. Soc. Am. B 10, 524–538 (1993). doi: 10.1364/JOSAB.10.000524.
https://doi.org/10.1364/JOSAB.10.000524
[50] H. M. Wiseman and G. J. Milburn. ``Quantum measurement and control''. Cambridge University Press. (2009). doi: 10.1017/CBO9780511813948.
https://doi.org/10.1017/CBO9780511813948
[51] F. Minganti, D. Huybrechts, C. Elouard, F. Nori, and I. I. Arkhipov. ``Creating and controlling exceptional points of non-hermitian hamiltonians via homodyne lindbladian invariance''. Phys. Rev. A 106, 042210 (2022). doi: 10.1103/PhysRevA.106.042210.
https://doi.org/10.1103/PhysRevA.106.042210
[52] S. Sorella, M. Casula, and D. Rocca. ``Weak binding between two aromatic rings: Feeling the van der Waals attraction by quantum Monte Carlo methods''. The Journal of Chemical Physics 127 (2007). doi: 10.1063/1.2746035.
https://doi.org/10.1063/1.2746035
[53] F. Vicentini, A. Biella, N. Regnault, and C. Ciuti. ``Variational neural-network ansatz for steady states in open quantum systems''. Phys. Rev. Lett. 122, 250503 (2019). doi: 10.1103/PhysRevLett.122.250503.
https://doi.org/10.1103/PhysRevLett.122.250503
[54] M. J. Hartmann and G. Carleo. ``Neural-network approach to dissipative quantum many-body dynamics''. Phys. Rev. Lett. 122, 250502 (2019). doi: 10.1103/PhysRevLett.122.250502.
https://doi.org/10.1103/PhysRevLett.122.250502
[55] A. Nagy and V. Savona. ``Variational quantum monte carlo method with a neural-network ansatz for open quantum systems''. Phys. Rev. Lett. 122, 250501 (2019). doi: 10.1103/PhysRevLett.122.250501.
https://doi.org/10.1103/PhysRevLett.122.250501
[56] M. Reh, M. Schmitt, and M. Gärttner. ``Time-dependent variational principle for open quantum systems with artificial neural networks''. Phys. Rev. Lett. 127, 230501 (2021). doi: 10.1103/PhysRevLett.127.230501.
https://doi.org/10.1103/PhysRevLett.127.230501
[57] I. Sutskever, J. Martens, G. Dahl, and G. Hinton. ``On the importance of initialization and momentum in deep learning''. In S. Dasgupta and D. McAllester, editors, Proceedings of the 30th International Conference on Machine Learning. Volume 28 of Proceedings of Machine Learning Research, pages 1139–1147. Atlanta, Georgia, USA (2013). PMLR. url: https://proceedings.mlr.press/v28/sutskever13.html.
https://proceedings.mlr.press/v28/sutskever13.html
[58] D. P. Kingma and J. Ba. ``Adam: A method for stochastic optimization'' (2017). arXiv:1412.6980.
arXiv:1412.6980
[59] S.-i. Amari. ``Natural Gradient Works Efficiently in Learning''. Neural Computation 10, 251–276 (1998). doi: 10.1162/089976698300017746.
https://doi.org/10.1162/089976698300017746
[60] S. Sorella. ``Wave function optimization in the variational monte carlo method''. Phys. Rev. B 71, 241103 (2005). doi: 10.1103/PhysRevB.71.241103.
https://doi.org/10.1103/PhysRevB.71.241103
[61] J. Yan and D. Bacon. ``The k-local pauli commuting hamiltonians problem is in p'' (2012). arXiv:1203.3906.
arXiv:1203.3906
[62] H. F. Trotter. ``On the product of semi-groups of operators''. In Proceedings of the American Mathematical Society. Volume 151, pages 545–551. (1959). doi: 10.1090/S0002-9939-1959-0108732-6.
https://doi.org/10.1090/S0002-9939-1959-0108732-6
[63] M. Suzuki. ``General theory of fractal path integrals with applications to many‐body theories and statistical physics''. Journal of Mathematical Physics 32, 400–407 (1991). doi: 10.1063/1.529425.
https://doi.org/10.1063/1.529425
[64] R. Y. Rubinstein and D. P. Kroese. ``Simulation and the monte carlo method''. John Wiley & Sons. (2016). doi: 10.1002/9781118631980.
https://doi.org/10.1002/9781118631980
[65] S. Mohamed, M. Rosca, M. Figurnov, and A. Mnih. ``Monte carlo gradient estimation in machine learning''. The Journal of Machine Learning Research 21, 5183–5244 (2020). url: http://jmlr.org/papers/v21/19-346.html.
http://jmlr.org/papers/v21/19-346.html
[66] D. Gottesman. ``The heisenberg representation of quantum computers'' (1998). arXiv:quant-ph/9807006.
arXiv:quant-ph/9807006
[67] D. Luo, Z. Chen, K. Hu, Z. Zhao, V. M. Hur, and B. K. Clark. ``Gauge-invariant and anyonic-symmetric autoregressive neural network for quantum lattice models''. Phys. Rev. Res. 5, 013216 (2023). doi: 10.1103/PhysRevResearch.5.013216.
https://doi.org/10.1103/PhysRevResearch.5.013216
[68] R. J. Elliott, P. Pfeuty, and C. Wood. ``Ising model with a transverse field''. Phys. Rev. Lett. 25, 443–446 (1970). doi: 10.1103/PhysRevLett.25.443.
https://doi.org/10.1103/PhysRevLett.25.443
[69] M. S. L. du Croo de Jongh and J. M. J. van Leeuwen. ``Critical behavior of the two-dimensional ising model in a transverse field: A density-matrix renormalization calculation''. Phys. Rev. B 57, 8494–8500 (1998). doi: 10.1103/PhysRevB.57.8494.
https://doi.org/10.1103/PhysRevB.57.8494
[70] H. Rieger and N. Kawashima. ``Application of a continuous time cluster algorithm to the two-dimensional random quantum ising ferromagnet''. The European Physical Journal B - Condensed Matter and Complex Systems 9, 233–236 (1999). doi: 10.1007/s100510050761.
https://doi.org/10.1007/s100510050761
[71] H. W. J. Blöte and Y. Deng. ``Cluster monte carlo simulation of the transverse ising model''. Phys. Rev. E 66, 066110 (2002). doi: 10.1103/PhysRevE.66.066110.
https://doi.org/10.1103/PhysRevE.66.066110
[72] A. F. Albuquerque, F. Alet, C. Sire, and S. Capponi. ``Quantum critical scaling of fidelity susceptibility''. Phys. Rev. B 81, 064418 (2010). doi: 10.1103/PhysRevB.81.064418.
https://doi.org/10.1103/PhysRevB.81.064418
[73] M. B. Hastings, I. González, A. B. Kallin, and R. G. Melko. ``Measuring renyi entanglement entropy in quantum monte carlo simulations''. Phys. Rev. Lett. 104, 157201 (2010). doi: 10.1103/PhysRevLett.104.157201.
https://doi.org/10.1103/PhysRevLett.104.157201
[74] D. N. Page. ``Average entropy of a subsystem''. Phys. Rev. Lett. 71, 1291–1294 (1993). doi: 10.1103/PhysRevLett.71.1291.
https://doi.org/10.1103/PhysRevLett.71.1291
[75] S. Bravyi, M. B. Hastings, and F. Verstraete. ``Lieb-robinson bounds and the generation of correlations and topological quantum order''. Physical review letters 97, 050401 (2006). doi: 10.1103/PhysRevLett.97.050401.
https://doi.org/10.1103/PhysRevLett.97.050401
[76] F. Vicentini et al. ``NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems''. SciPost Phys. CodebasesPage 7 (2022). doi: 10.21468/SciPostPhysCodeb.7.
https://doi.org/10.21468/SciPostPhysCodeb.7
[77] G. Carleo et al. ``NetKet: A machine learning toolkit for many-body quantum systems''. SoftwareXPage 100311 (2019). doi: 10.1016/j.softx.2019.100311.
https://doi.org/10.1016/j.softx.2019.100311
[78] D. Häfner and F. Vicentini. ``mpi4jax: Zero-copy MPI communication of JAX arrays''. Journal of Open Source Software 6 (2021). doi: 10.21105/joss.03419.
https://doi.org/10.21105/joss.03419
[79] A. Sinibaldi and F. Vicentini. ``netket_fidelity package, v0.0.2''. doi: 10.5281/zenodo.8344170.
https://doi.org/10.5281/zenodo.8344170
[80] V. Havlicek. ``Amplitude Ratios and Neural Network Quantum States''. Quantum 7, 938 (2023). doi: 10.22331/q-2023-03-02-938.
https://doi.org/10.22331/q-2023-03-02-938
Cited by
[1] 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 1, 9404 (2024).
[2] Zhou-Quan Wan and Shiwei Zhang, "Addressing the infinite variance problem in fermionic Monte Carlo simulations: Retrospective error remediation and the exact bridge link method", Physical Review E 112 6, 065309 (2025).
[3] Liam Bernheimer, Hristiana Atanasova, and Guy Cohen, "Determinant- and derivative-free quantum Monte Carlo within the stochastic representation of wavefunctions", Reports on Progress in Physics 87 11, 118001 (2024).
[4] Joseph Vovrosh, Sergi Julià-Farré, Wladislaw Krinitsin, Michael Kaicher, Fergus Hayes, Emmanuel Gottlob, Augustine Kshetrimayum, Kemal Bidzhiev, Simon B. Jäger, Markus Schmitt, Joseph Tindall, Constantin Dalyac, Tiago Mendes-Santos, and Alexandre Dauphin, "Simulating dynamics of the two-dimensional transverse-field Ising model: A comparative study of large-scale classical numerics", Physical Review Research 8 2, 023311 (2026).
[5] Joseph Tindall, Antonio Francesco Mello, Matthew Fishman, E. Miles Stoudenmire, and Dries Sels, "Dynamics of disordered quantum systems with two- and three-dimensional tensor networks", Science 392 6800, 868 (2026).
[6] Zhou-Quan Wan, Roeland Wiersema, and Shiwei Zhang, "Removing Nodal and Support-Mismatch Pathologies in Variational Monte Carlo via Blurred Sampling", Physical Review X 16 3, 031059 (2026).
[7] 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).
[8] Tatiana Vovk, Anka Van de Walle, Hannes Pichler, and Annabelle Bohrdt, "Neural quantum states for emitter dynamics in waveguide QED", Physical Review A 113 5, 053708 (2026).
[9] Wladislaw Krinitsin, Niklas Tausendpfund, Markus Heyl, Matteo Rizzi, and Markus Schmitt, "Time evolution of the quantum Ising model in two dimensions using tree tensor networks", Physical Review B 112 13, 134310 (2025).
[10] Adrien Kahn, Luca Gravina, and Filippo Vicentini, "Variational subspace methods and application to improving variational Monte Carlo dynamics", Quantum 10, 2082 (2026).
[11] 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).
[12] David Linteau, Saverio Moroni, Giuseppe Carleo, and Markus Holzmann, "Variance reduction for forces and pressure in variational Monte Carlo", The Journal of Chemical Physics 165 13, 134106 (2026).
[13] Linda Mauron, Zakari Denis, Jannes Nys, and Giuseppe Carleo, "Predicting topological entanglement entropy in a Rydberg analogue simulator", Nature Physics 21 8, 1332 (2025).
[14] 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).
[15] Brandon Barton, Juan Carrasquilla, Christopher Roth, and Agnes Valenti, "Connectivity determines the capability of sparse neural network quantum states", Machine Learning: Science and Technology 7 3, 035025 (2026).
[16] Matija Medvidović, Alev Orfi, Juan Carrasquilla, and Dries Sels, "Adiabatic Transport of Neural Network Quantum States", PRX Intelligence 1 1, 013020 (2026).
[17] Andrew D. King, Alberto Nocera, Marek M. Rams, Jacek Dziarmaga, Roeland Wiersema, William Bernoudy, Jack Raymond, Nitin Kaushal, Niclas Heinsdorf, Richard Harris, Kelly Boothby, Fabio Altomare, Mohsen Asad, Andrew J. Berkley, Martin Boschnak, Kevin Chern, Holly Christiani, Samantha Cibere, Jake Connor, Martin H. Dehn, Rahul Deshpande, Sara Ejtemaee, Pau Farre, Kelsey Hamer, Emile Hoskinson, Shuiyuan Huang, Mark W. Johnson, Samuel Kortas, Eric Ladizinsky, Trevor Lanting, Tony Lai, Ryan Li, Allison J. R. MacDonald, Gaelen Marsden, Catherine C. McGeoch, Reza Molavi, Travis Oh, Richard Neufeld, Mana Norouzpour, Joel Pasvolsky, Patrick Poitras, Gabriel Poulin-Lamarre, Thomas Prescott, Mauricio Reis, Chris Rich, Mohammad Samani, Benjamin Sheldan, Anatoly Smirnov, Edward Sterpka, Berta Trullas Clavera, Nicholas Tsai, Mark Volkmann, Alexander M. Whiticar, Jed D. Whittaker, Warren Wilkinson, Jason Yao, T. J. Yi, Anders W. Sandvik, Gonzalo Alvarez, Roger G. Melko, Juan Carrasquilla, Marcel Franz, and Mohammad H. Amin, "Beyond-classical computation in quantum simulation", Science 388 6743, 199 (2025).
[18] Yaofeng Chen and Li You, "Optimal Control of Unknown Collective Spin Systems via a Neural Network Surrogate", Chinese Physics Letters 42 10, 100601 (2025).
[19] Matthew Kiser, Matthias Beuerle, and Fedor Šimkovic, "Contextual Subspace Auxiliary-Field Quantum Monte Carlo: Improved Bias with Reduced Quantum Resources", Journal of Chemical Theory and Computation 21 5, 2256 (2025).
[20] 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).
[21] Douglas Hendry, Alessandro Sinibaldi, and Giuseppe Carleo, "Grassmann variational Monte Carlo with neural wave functions", Physical Review B 114 13, 134404 (2026).
[22] 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 6, 211 (2024).
[23] Zechen Tang, Haoxiang Chen, Yang Li, Yubing Qian, Yuxiang Wang, Weizhong Fu, Jialin Li, Chen Si, Wenhui Duan, Ji Chen, and Yong Xu, "Deep-learning electronic structure calculations", Nature Computational Science 5 12, 1133 (2025).
[24] Yantao Wu and Wen-Yuan Liu, "Accurate Gauge-Invariant Tensor-Network Simulations for Abelian Lattice Gauge Theory in (2+1)D : Ground-State and Real-Time Dynamics", Physical Review Letters 135 13, 130401 (2025).
[25] 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).
[26] Wenxuan Zhang, Bo Xing, Xiansong Xu, and Dario Poletti, "Paths towards time evolution with larger neural-network quantum states", Computer Physics Communications 312, 109577 (2025).
[27] Eimantas Ledinauskas and Egidijus Anisimovas, "Scalable imaginary time evolution with neural network quantum states", SciPost Physics 15 6, 229 (2023).
[28] Tim S Hindges, Wenhao He, and Ju Li, "Recent advances in electronic structure learning", Modelling and Simulation in Materials Science and Engineering 34 5, 053001 (2026).
[29] 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).
[30] Tarun Advaith Kumar, Leon Balents, Timothy H. Hsieh, and Roger G. Melko, "Autoregressive typical thermal states", Annals of Physics 487, 170362 (2026).
[31] Hannah Lange, Fabian Döschl, Juan Carrasquilla, and Annabelle Bohrdt, "Neural network approach to quasiparticle dispersions in doped antiferromagnets", Communications Physics 7 1, 187 (2024).
[32] Sara Santos, Xinyu Song, and Vincenzo Savona, "Low-Rank Variational Quantum Algorithm for the Dynamics of Open Quantum Systems", Quantum 9, 1620 (2025).
[33] 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).
[34] Yantao Wu and Jannes Nys, "Real-Time Dynamics in Two Dimensions with Tensor Network States via Time-Dependent Variational Monte Carlo Method", PRX Quantum 7 3, 033035 (2026).
[35] Jannes Nys, Zakari Denis, and Giuseppe Carleo, "Real-time quantum dynamics of thermal states with neural thermofields", Physical Review B 109 23, 235120 (2024).
[36] Ashish Joshi, Robert Peters, and Thore Posske, "Quantum skyrmion dynamics studied by neural network quantum states", Physical Review B 110 10, 104411 (2024).
[37] Ao Chen, Vighnesh Dattatraya Naik, and Markus Heyl, "Convolutional transformer wave functions", Physical Review Research 8 2, L022040 (2026).
[38] Haimeng Zhao, Giuseppe Carleo, and Filippo Vicentini, "Empirical Sample Complexity of Neural Network Mixed State Reconstruction", Quantum 8, 1358 (2024).
[39] 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).
[40] Alessandro Sinibaldi, Antonio Francesco Mello, Mario Collura, and Giuseppe Carleo, "Nonstabilizerness of neural quantum states", Physical Review Research 7 4, 043289 (2025).
[41] Matteo Acciai, Liliana Arrachea, and Janine Splettstoesser, "Quantum transport phenomena induced by time-dependent fields", La Rivista del Nuovo Cimento 48 10, 653 (2025).
[42] Massimo Bortone, Yannic Rath, and George H. Booth, "Impact of conditional modelling for a universal autoregressive quantum state", Quantum 8, 1245 (2024).
[43] Gian Gentinetta, Friederike Metz, and Giuseppe Carleo, "Correcting and Extending Trotterized Quantum Many-Body Dynamics", PRX Quantum 6 3, 030361 (2025).
[44] 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).
[45] Johannes Christmann, Petr Ivashkov, Mattia Chiurco, and Guglielmo Mazzola, "From quantum-enhanced to quantum-inspired Monte Carlo", Physical Review A 111 4, 042615 (2025).
[46] Dennis Wagner, Andreas Klümper, and Jesko Sirker, "Thermodynamics based on neural networks", Physical Review B 109 15, 155128 (2024).
[47] Shaojun Gui, Tak-San Ho, and Herschel Rabitz, "Discrete real-time learning of quantum-state subspace evolution of many-body systems in the presence of time-dependent control fields", Physical Review A 110 5, 052412 (2024).
[48] Joseph Vovrosh, Tiago Mendes-Santos, Hadriel Mamann, Kemal Bidzhiev, Fergus Hayes, Bruno Ximenez, Lucas Béguin, Constantin Dalyac, and Alexandre Dauphin, "Resource Assessment of Classical and Quantum Hardware for Post-Quench Dynamics", PRX Energy 5 3, 033007 (2026).
[49] Zhuo Chen and Di Luo, 2024 IEEE 6th International Conference on Trust, Privacy and Security in Intelligent Systems, and Applications (TPS-ISA) 516 (2024) ISBN:979-8-3503-8674-5.
[50] Luca Gravina, Vincenzo Savona, and Filippo Vicentini, "Neural Projected Quantum Dynamics: a systematic study", Quantum 9, 1803 (2025).
[51] 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", Physical Review Letters 137 9, 093402 (2026).
[52] Matija Medvidović and Javier Robledo Moreno, "Neural-network quantum states for many-body physics", The European Physical Journal Plus 139 7, 631 (2024).
[53] David Linteau, Gabriel Pescia, Jannes Nys, Giuseppe Carleo, and Markus Holzmann, "Phase Diagram and Crystal Melting of Helium-4 in Two Dimensions", Physical Review Letters 134 24, 246001 (2025).
[54] Alessandro Sinibaldi, Douglas Hendry, Filippo Vicentini, and Giuseppe Carleo, "Time-Dependent Neural Galerkin Method for Quantum Dynamics", Physical Review Letters 136 12, 120402 (2026).
[55] 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).
[56] Hannah Lange, Anka Van de Walle, Atiye Abedinnia, and Annabelle Bohrdt, "From architectures to applications: a review of neural quantum states", Quantum Science and Technology 9 4, 040501 (2024).
[57] Anna Dawid, Julian Arnold, Borja Requena, Alexander Gresch, Marcin Płodzień, Kaelan Donatella, Kim A. Nicoli, Paolo Stornati, Rouven Koch, Miriam Büttner, Robert Okuła, Gorka Muñoz-Gil, Rodrigo A. Vargas-Hernández, Alba Cervera-Lierta, Juan Carrasquilla, Vedran Dunjko, Marylou Gabrié, Patrick Huembeli, Evert van Nieuwenburg, Filippo Vicentini, Lei Wang, Sebastian J. Wetzel, Giuseppe Carleo, Eliška Greplová, Roman Krems, Florian Marquardt, Michał Tomza, Maciej Lewenstein, and Alexandre Dauphin, "Modern applications of machine learning in quantum sciences", arXiv:2204.04198, (2022).
[58] 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).
[59] Tatiana Vovk, Anka Van de Walle, Hannes Pichler, and Annabelle Bohrdt, "Neural quantum states for emitter dynamics in waveguide QED", arXiv:2508.08964, (2025).
[60] Thomas Spriggs, Eliska Greplova, Juan Carrasquilla, and Jannes Nys, "Accurate ground states of $SU(2)$ lattice gauge theory in 2+1D and 3+1D", arXiv:2509.12323, (2025).
[61] Elena Orlova, Aleksei Ustimenko, Ruoxi Jiang, Peter Y. Lu, and Rebecca Willett, "Deep Stochastic Mechanics", arXiv:2305.19685, (2023).
[62] Stefano Barison, Filippo Vicentini, and Giuseppe Carleo, "Variational Embeddings for Many Body Quantum Systems", arXiv:2309.08666, (2023).
[63] Xiu-Zhe Luo, Di Luo, and Roger G. Melko, "Operator Learning Renormalization Group", arXiv:2403.03199, (2024).
[64] Julien Gacon, "Scalable Quantum Algorithms for Noisy Quantum Computers", arXiv:2403.00940, (2024).
[65] Debbie Eeltink, Filippo Vicentini, and Vincenzo Savona, "Variational dynamics of open quantum systems in phase space", arXiv:2307.07429, (2023).
[66] Zhuo Chen, Jacob McCarran, Esteban Vizcaino, Marin Soljačić, and Di Luo, "TENG: Time-Evolving Natural Gradient for Solving PDEs With Deep Neural Nets Toward Machine Precision", arXiv:2404.10771, (2024).
[67] Alessandro Lovato, Giuseppe Carleo, Bryce Fore, Morten Hjorth-Jensen, Jane Kim, Arnau Rios, and Noemi Rocco, "Neural-network quantum states for the nuclear many-body problem", arXiv:2602.13826, (2026).
[68] Riccardo Rende, Alessandro Sinibaldi, Luciano Loris Viteritti, Roeland Wiersema, Antoine Georges, and Giuseppe Carleo, "Scaling Laws for Neural-Network Quantum States", arXiv:2606.02794, (2026).
[69] Hrvoje Vrcan and Johan H. Mentink, "Instability of explicit time integration for strongly quenched dynamics with neural quantum states", arXiv:2507.17421, (2025).
[70] Yantao Wu and Jannes Nys, "Real-Time Dynamics in Two Dimensions with Tensor Network States via Time-Dependent Variational Monte Carlo", arXiv:2512.06768, (2025).
[71] Alessandro Santini, Stefano Barison, and Filippo Vicentini, "Classically Prepared, Quantumly Evolved: Hybrid Algorithm for Molecular Spectra", arXiv:2510.24911, (2025).
[72] 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).
[73] 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).
[74] Wladislaw Krinitsin and Markus Schmitt, "Time-dependent variational Monte Carlo without bias", arXiv:2605.03930, (2026).
[75] Alessandro Sinibaldi, Luciano Loris Viteritti, Riccardo Rende, Fakher F. Assaad, and Giuseppe Carleo, "Thermalization Dynamics in the Two-Dimensional Hubbard Model with Neural-Network Quantum States", arXiv:2606.05293, (2026).
[76] Jonas B. Rigo, Björn J. Wurst, Rajah Nutakki, Markus Schmitt, and Dante Kennes, "Neural quantum states in condensed matter: advances, best practices, and prospects", arXiv:2608.21291, (2026).
[77] Linda Mauron, Luciano Loris Viteritti, Zakari Denis, Riccardo Rende, and Giuseppe Carleo, "Kibble-Zurek Dynamics in Two-dimensional Frustrated Systems with a Neural Foundation-state Subspace Method", arXiv:2609.05556, (2026).
[78] Adil Attar, Amine M. Aboussalah, and Mohamed Hibat-Allah, "Quantum Geometric Tensor Preconditioning for Stable Training of Recurrent Neural Quantum States", arXiv:2608.18065, (2026).
[79] Bharadwaj Chowdary Mummaneni and Bo Xing, "Neural Quantum States for Nuclear Magnetic Resonance Spectroscopy", arXiv:2608.00178, (2026).
[80] Lexin Ding and Markus Reiher, "Propagating sparsely supported states with time-dependent neural quantum states", arXiv:2609.06054, (2026).
[81] Yi-Ran Xue, Rui Wang, Baigeng Wang, and Chenan Wei, "Gradient-estimator design overcomes trainability barriers in neural-network-based variational optimization", arXiv:2609.22342.
[82] Wladislaw Krinitsin, Jonas B. Rigo, Mohammad Abedi, and Markus Schmitt, "Geometric inflation of deviations challenges neural quantum states in dynamics of quantum Ising models", arXiv:2609.07645, (2026).
The above citations are from Crossref's cited-by service (last updated successfully 2026-11-03 20:12:27) and SAO/NASA ADS (last updated successfully 2026-11-03 20:12:29). 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.