Unbiasing time-dependent Variational Monte Carlo by projected quantum evolution

Alessandro Sinibaldi1,2, Clemens Giuliani1,2, Giuseppe Carleo1,2, and Filippo Vicentini1,2,3

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

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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.

Monte Carlo variational methods are a powerful class of numerical techniques to simulate many-body quantum systems. While these algorithms have delivered state-of-the-art results for ground state search of frustrated and high-dimensional problems, variational calculations have yet to provide significant improvements over existing approaches for time evolution. Variational methods to simulate the quantum dynamics on classical or quantum hardware either integrate an explicit system of equations for the wave function parameters, the so-called time-dependent Variational Principle (TDVP), or implicitly solve a time-step optimization problem.
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.

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[1] I. M. Georgescu, S. Ashhab, and F. Nori. ``Quantum simulation''. Rev. Mod. Phys. 86, 153–185 (2014). doi: 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.

[3] M. A. Nielsen and I. L. Chuang. ``Quantum computation and quantum information: 10th anniversary edition''. Cambridge University Press. (2010). doi: 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15] S. R. White. ``Density matrix formulation for quantum renormalization groups''. Phys. Rev. Lett. 69, 2863–2866 (1992). doi: 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.

[17] D. Ceperley and B. Alder. ``Quantum monte carlo''. Science 231, 555–560 (1986). doi: 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27] D. Wu et al. ``Variational benchmarks for quantum many-body problems'' (2023) 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.

[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.

[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.

[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.

[32] B. Jónsson, B. Bauer, and G. Carleo. ``Neural-network states for the classical simulation of quantum computing'' (2018). 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.

[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.

[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.

[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.

[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.

[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.

[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.

[40] M. V. den Nest. ``Simulating quantum computers with probabilistic methods'' (2010). arXiv:0911.1624.

[41] N. Moiseyev. ``Non-hermitian quantum mechanics''. Cambridge University Press. (2011). doi: 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.

[43] A. McLachlan. ``A variational solution of the time-dependent schrodinger equation''. Molecular Physics 8, 39–44 (1964). doi: 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.

[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.

[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.

[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.

[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.

[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.

[50] H. M. Wiseman and G. J. Milburn. ``Quantum measurement and control''. Cambridge University Press. (2009). doi: 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.

[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.

[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.

[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.

[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.

[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.

[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.

[58] D. P. Kingma and J. Ba. ``Adam: A method for stochastic optimization'' (2017). arXiv:1412.6980.

[59] S.-i. Amari. ``Natural Gradient Works Efficiently in Learning''. Neural Computation 10, 251–276 (1998). doi: 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.

[61] J. Yan and D. Bacon. ``The k-local pauli commuting hamiltonians problem is in p'' (2012). 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.

[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.

[64] R. Y. Rubinstein and D. P. Kroese. ``Simulation and the monte carlo method''. John Wiley & Sons. (2016). doi: 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.

[66] D. Gottesman. ``The heisenberg representation of quantum computers'' (1998). 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.

[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.

[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.

[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.

[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.

[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.

[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.

[74] D. N. Page. ``Average entropy of a subsystem''. Phys. Rev. Lett. 71, 1291–1294 (1993). doi: 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.

[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.

[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.

[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.

[79] A. Sinibaldi and F. Vicentini. ``netket_fidelity package, v0.0.2''. doi: 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.

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