- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Removing Nodal and Support-Mismatch Pathologies in Variational Monte Carlo via Blurred Sampling

Zhou-Quan Wan*,†, Roeland Wiersema*,‡, and Shiwei Zhang§

  • Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA

  • *These authors contributed equally to this work.
  • †Contact author: zwan@flatironinstitute.org
  • ‡Contact author: rwiersema@flatironinstitute.org
  • §Contact author: szhang@flatironinstitute.org
PDF

Phys. Rev. X 16, 031059 – Published 2 September, 2026

DOI: https://doi.org/10.1103/jrn5-gv19

Abstract

Variational Monte Carlo (VMC) is a powerful and fast-growing method for optimizing and evolving parametrized many-body wave functions, especially with modern neural-network quantum states. In practice, however, the stochastic estimators that form the backbone of the method can become unstable or biased due to the presence of nodes, a ubiquitous feature of quantum wave functions. In the continuum, this results in heavy-tailed estimators with potentially divergent variances, while in discrete Hilbert spaces the sampling distribution can miss parts of the support needed to form unbiased estimators. These statistical pathologies lead to unreliable optimization trajectories in stochastic reconfiguration or incorrect variational dynamics in time-dependent variational Monte Carlo (t-VMC) and severely limit the power of the numerical simulations. We introduce blurred sampling to address these difficulties. The method has a number of rigorous properties that make it well behaved, effective, and computationally efficient. Additionally, it is a postprocessing approach that can be used without modifying the underlying sampler and incurs only minimal overhead. We demonstrate its effectiveness in a broad range of problems—including ground-state optimization in both continuum and discrete settings as well as in time-dependent variational Monte Carlo, where it resolves previously identified failure modes—and further apply it to large-scale problems in spin dynamics. This work establishes a broadly applicable framework for robust VMC and t-VMC calculations.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Variational Monte Carlo simulations of quantum many-body systems suffer from fundamental statistical instabilities, including infinite variance and systematic estimator bias. We resolved these computational issues by developing blurred sampling, a Monte Carlo postprocessing method that eliminates diverging fluctuations and missing estimator contributions.

By smoothing statistical singularities without sacrificing computational efficiency, our approach maintains compatibility with established stochastic algorithms while stabilizing simulations in regimes where standard methods fail. Our work establishes a robust postprocessing protocol for quantum many-body calculations, extending the precision of variational simulations across physics and chemistry.

Article Text

References (100)

  1. W. L. McMillan, Ground state of liquid He4, Phys. Rev. 138, A442 (1965).
  2. D. M. Ceperley and B. J. Alder, Ground state of the electron gas by a stochastic method, Phys. Rev. Lett. 45, 566 (1980).
  3. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  4. K. Choo, G. Carleo, N. Regnault, and T. Neupert, Symmetries and many-body excitations with neural-network quantum states, Phys. Rev. Lett. 121, 167204 (2018).
  5. S. Pathak, B. Busemeyer, J. N. B. Rodrigues, and L. K. Wagner, Excited states in variational Monte Carlo using a penalty method, J. Chem. Phys. 154, 034101 (2021).
  6. M. T. Entwistle, Z. Schätzle, P. A. Erdman, J. Hermann, and F. Noé, Electronic excited states in deep variational Monte Carlo, Nat. Commun. 14, 274 (2023).
  7. H. Lange, F. Döschl, J. Carrasquilla, and A. Bohrdt, Neural network approach to quasiparticle dispersions in doped antiferromagnets, Commun. Phys. 7, 187 (2024).
  8. 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(R) (2014).
  9. M. Schmitt and M. Heyl, Quantum many-body dynamics in two dimensions with artificial neural networks, Phys. Rev. Lett. 125, 100503 (2020).
  10. I. L. Gutiérrez and C. B. Mendl, Real time evolution with neural-network quantum states, Quantum 6, 627 (2022).
  11. 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, Sci. Adv. 8, eabl6850 (2022).
  12. M. Schmitt and M. Heyl, Simulating dynamics of correlated matter with neural quantum states, arXiv:2506.03124.
  13. S. Sorella, Green function Monte Carlo with stochastic reconfiguration, Phys. Rev. Lett. 80, 4558 (1998).
  14. N. Irikura and H. Saito, Neural-network quantum states at finite temperature, Phys. Rev. Res. 2, 013284 (2020).
  15. H. Xie, Z.-H. Li, H. Wang, L. Zhang, and L. Wang, Deep variational free energy approach to dense hydrogen, Phys. Rev. Lett. 131, 126501 (2023).
  16. J. Nys, Z. Denis, and G. Carleo, Real-time quantum dynamics of thermal states with neural thermofields, Phys. Rev. B 109, 235120 (2024).
  17. J. Nys and J. Carrasquilla, Fermionic neural Gibbs states, arXiv:2512.04663.
  18. D. Luo and B. K. Clark, Backflow transformations via neural networks for quantum many-body wave functions, Phys. Rev. Lett. 122, 226401 (2019).
  19. Y. Nomura and M. Imada, Dirac-type nodal spin liquid revealed by refined quantum many-body solver using neural-network wave function, correlation ratio, and level spectroscopy, Phys. Rev. X 11, 031034 (2021).
  20. W.-Y. Liu, Y.-Z. Huang, S.-S. Gong, and Z.-C. Gu, Accurate simulation for finite projected entangled pair states in two dimensions, Phys. Rev. B 103, 235155 (2021).
  21. J. R. Moreno, G. Carleo, A. Georges, and J. Stokes, Fermionic wave functions from neural-network constrained hidden states, Proc. Natl. Acad. Sci. U.S.A. 119, e2122059119 (2022).
  22. M. Medvidović and J. R. Moreno, Neural-network quantum states for many-body physics, Eur. Phys. J. Plus 139, 1 (2024).
  23. A. Chen and M. Heyl, Empowering deep neural quantum states through efficient optimization, Nat. Phys. 20, 1476 (2024).
  24. H. Lange, A. Van de Walle, A. Abedinnia, and A. Bohrdt, From architectures to applications: A review of neural quantum states, Quantum Sci. Technol. 9, 040501 (2024).
  25. Y. Gu, W. Li, H. Lin, B. Zhan, R. Li, Y. Huang, D. He, Y. Wu, T. Xiang, M. Qin, L. Wang, and D. Lv, Solving the hubbard model with neural quantum states, arXiv:2507.02644.
  26. W.-Y. Liu, H. Zhai, R. Peng, Z.-C. Gu, and G. K.-L. Chan, Accurate simulation of the Hubbard model with finite fermionic projected entangled pair states, Phys. Rev. Lett. 134, 256502 (2025).
  27. A. Chen, Z.-Q. Wan, A. Sengupta, A. Georges, and C. Roth, Neural network-augmented Pfaffian wave-functions for scalable simulations of interacting fermions, arXiv:2507.10705.
  28. X. Li, Y. Chen, B. Li, H. Chen, F. Wu, J. Chen, and W. Ren, Deep learning sheds light on integer and fractional topological insulators, arXiv:2503.11756.
  29. L. L. Viteritti, R. Rende, S. Sachdev, and G. Carleo, Approaching the thermodynamic limit with neural-network quantum states, arXiv:2602.02665.
  30. J. Hermann, Z. Schätzle, and F. Noé, Deep-neural-network solution of the electronic Schrödinger equation, Nat. Chem. 12, 891 (2020).
  31. 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. Res. 2, 033429 (2020).
  32. K. Choo, A. Mezzacapo, and G. Carleo, Fermionic neural-network states for ab-initio electronic structure, Nat. Commun. 11, 2368 (2020).
  33. J. Hermann, J. Spencer, K. Choo, A. Mezzacapo, W. M. C. Foulkes, D. Pfau, G. Carleo, and F. Noé, Ab initio quantum chemistry with neural-network wavefunctions, Nat. Rev. Chem. 7, 692 (2023).
  34. I. von Glehn, J. S. Spencer, and D. Pfau, A self-attention ansatz for ab-initio quantum chemistry, arXiv:2211.13672.
  35. H. Shang, C. Guo, Y. Wu, Z. Li, and J. Yang, Solving Schrödinger equation with a language model, arXiv:2307.09343.
  36. D. Pfau, S. Axelrod, H. Sutterud, I. von Glehn, and J. S. Spencer, Accurate computation of quantum excited states with neural networks, Science 385, eadn0137 (2024).
  37. J. Kim, G. Pescia, B. Fore, J. Nys, G. Carleo, S. Gandolfi, M. Hjorth-Jensen, and A. Lovato, Neural-network quantum states for ultra-cold Fermi gases, Commun. Phys. 7, 148 (2024).
  38. M. Wilson, S. Moroni, M. Holzmann, N. Gao, F. Wudarski, T. Vegge, and A. Bhowmik, Neural network ansatz for periodic wave functions and the homogeneous electron gas, Phys. Rev. B 107, 235139 (2023).
  39. G. Cassella, H. Sutterud, S. Azadi, N. D. Drummond, D. Pfau, J. S. Spencer, and W. M. C. Foulkes, Discovering quantum phase transitions with fermionic neural networks, Phys. Rev. Lett. 130, 036401 (2023).
  40. C. Smith, Y. Chen, R. Levy, Y. Yang, M. A. Morales, and S. Zhang, Unified variational approach description of ground-state phases of the two-dimensional electron gas, Phys. Rev. Lett. 133, 266504 (2024).
  41. G. Pescia, J. Nys, J. Kim, A. Lovato, and G. Carleo, Message-passing neural quantum states for the homogeneous electron gas, Phys. Rev. B 110, 035108 (2024).
  42. M. Geier, K. Nazaryan, T. Zaklama, and L. Fu, Self-attention neural network for solving correlated electron problems in solids, Phys. Rev. B 112, 045119 (2025).
  43. Y. Teng, D. D. Dai, and L. Fu, Solving the fractional quantum Hall problem with self-attention neural network, Phys. Rev. B 111, 205117 (2025).
  44. Y. Qian, T. Zhao, J. Zhang, T. Xiang, X. Li, and J. Chen, Describing Landau level mixing in fractional quantum Hall states with deep learning, Phys. Rev. Lett. 134, 176503 (2025).
  45. C.-T. Li, T. Ong, M. Geier, H. Lin, and L. Fu, Attention is all you need to solve chiral superconductivity, arXiv:2509.03683.
  46. Z. Ge, C. Smith, Z. He, Y. Yang, Q. Li, H.-L. Kim, Z. Xiang, J. Xiao, W. Zhou, S. Kahn et al., Visualizing the impact of quenched disorder on 2D electron Wigner solids, Nature (London) 654, 902 (2026).
  47. C. Smith, Y. Yang, Z.-Q. Wan, Y. Chen, M. A. Morales, and S. Zhang, Neural network discovery of paired Wigner crystals in artificial graphene, arXiv:2602.16798.
  48. M. Medvidović and G. Carleo, Classical variational simulation of the quantum approximate optimization algorithm, npj Quantum Inf. 7, 101 (2021).
  49. A. D. King, A. Nocera, M. M. Rams, J. Dziarmaga, R. Wiersema, W. Bernoudy, J. Raymond, N. Kaushal, N. Heinsdorf, R. Harris et al., Beyond-classical computation in quantum simulation, Science 388, 199 (2025).
  50. R. Haghshenas, E. Chertkov, M. Mills, W. Kadow, S.-H. Lin, Y.-H. Chen, C. Cade, I. Niesen, T. Begusić, M. S. Rudolph et al., Digital quantum magnetism on a trapped-ion quantum computer, Nature (London) 653, 56 (2026).
  51. S. Sorella, Generalized Lanczos algorithm for variational quantum Monte Carlo, Phys. Rev. B 64, 024512 (2001).
  52. J. R. Trail, Heavy-tailed random error in quantum Monte Carlo, Phys. Rev. E 77, 016703 (2008).
  53. H. Shi and S. Zhang, Infinite variance in fermion quantum Monte Carlo calculations, Phys. Rev. E 93, 033303 (2016).
  54. P. L. Ríos and G. J. Conduit, Tail-regression estimator for heavy-tailed distributions of known tail indices and its application to continuum quantum Monte Carlo data, Phys. Rev. E 99, 063312 (2019).
  55. Z.-Q. Wan and S. Zhang, Addressing the infinite variance problem in fermionic Monte Carlo simulations: Retrospective error remediation and the exact bridge link method, Phys. Rev. E 112, 065309 (2025).
  56. A. Sinibaldi, C. Giuliani, G. Carleo, and F. Vicentini, Unbiasing time-dependent variational Monte Carlo by projected quantum evolution, Quantum 7, 1131 (2023).
  57. D. Hofmann, G. Fabiani, J. H. Mentink, G. Carleo, and M. A. Sentef, Role of stochastic noise and generalization error in the time propagation of neural-network quantum states, SciPost Phys. 12, 165 (2022).
  58. S. Pathak and L. K. Wagner, A light weight regularization for wave function parameter gradients in quantum Monte Carlo, AIP Adv. 10, 085213 (2020).
  59. L. Gravina, V. Savona, and F. Vicentini, Neural projected quantum dynamics: A systematic study, Quantum 9, 1803 (2025).
  60. D. Linteau, S. Moroni, G. Carleo, and M. Holzmann, Variance reduction for forces and pressure in variational Monte Carlo, arXiv:2603.14521.
  61. P. J. Robinson, S. D. Pineda Flores, and E. Neuscamman, Excitation variance matching with limited configuration interaction expansions in variational Monte Carlo, J. Chem. Phys. 147, 164114 (2017).
  62. L. Otis, I. M. Craig, and E. Neuscamman, A hybrid approach to excited-state-specific variational Monte Carlo and doubly excited states, J. Chem. Phys. 153, 234105 (2020).
  63. K. Inui, Y. Kato, and Y. Motome, Determinant-free fermionic wave function using feed-forward neural networks, Phys. Rev. Res. 3, 043126 (2021).
  64. A. Misery, L. Gravina, A. Santini, and F. Vicentini, Looking elsewhere: Improving variational Monte Carlo gradients by importance sampling, Mach. Learn. 7, 015035 (2026).
  65. C. Attaccalite and S. Sorella, Stable liquid hydrogen at high pressure by a novel ab initio molecular-dynamics calculation, Phys. Rev. Lett. 100, 114501 (2008).
  66. J. R. Trail, Alternative sampling for variational quantum Monte Carlo, Phys. Rev. E 77, 016704 (2008).
  67. A. Malyshev, M. Schmitt, and A. I. Lvovsky, Neural quantum states and peaked molecular wave functions: Curse or blessing?, arXiv:2408.07625.
  68. A.-J. Liu and B. K. Clark, Efficient optimization of neural network backflow for ab initio quantum chemistry, Phys. Rev. B 112, 155162 (2025).
  69. B. Uria, M.-A. Côté, K. Gregor, I. Murray, and H. Larochelle, Neural autoregressive distribution estimation, J. Mach. Learn. Res. 17, 1 (2016), http://jmlr.org/papers/v17/16-272.html.
  70. M. Hibat-Allah, M. Ganahl, L. E. Hayward, R. G. Melko, and J. Carrasquilla, Recurrent neural network wave functions, Phys. Rev. Res. 2, 023358 (2020).
  71. 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).
  72. S. Humeniuk, Y. Wan, and L. Wang, Autoregressive neural Slater-Jastrow ansatz for variational Monte Carlo simulation, SciPost Phys. 14, 171 (2023).
  73. K. Sprague and S. Czischek, Variational Monte Carlo with large patched transformers, Commun. Phys. 7, 90 (2024).
  74. M. S. Moss, R. Wiersema, M. Hibat-Allah, J. Carrasquilla, and R. G. Melko, Leveraging recurrence in neural network wavefunctions for large-scale simulations of heisenberg antiferromagnets on the square lattice, Phys. Rev. B 112, 134450 (2025).
  75. M. S. Moss, R. Wiersema, M. Hibat-Allah, J. Carrasquilla, and R. G. Melko, Leveraging recurrence in neural network wavefunctions for large-scale simulations of heisenberg antiferromagnets on the triangular lattice, Phys. Rev. B 112, 134449 (2025).
  76. G. Carleo, F. Becca, M. Schiró, and M. Fabrizio, Localization and glassy dynamics of many-body quantum systems, Sci. Rep. 2, 243 (2012).
  77. D. M. Ceperley, Fermion nodes, J. Stat. Phys. 63, 1237 (1991).
  78. W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Quantum Monte Carlo simulations of solids, Rev. Mod. Phys. 73, 33 (2001).
  79. L. Mitas, Structure of fermion nodes and nodal cells, Phys. Rev. Lett. 96, 240402 (2006).
  80. R. J. Needs, M. D. Towler, N. D. Drummond, and P. L. Ríos, Continuum variational and diffusion quantum Monte Carlo calculations, J. Phys. Condens. Matter 22, 023201 (2009).
  81. P. Lévy, Calcul des probabilités (Gauthier-Villars, 1925).
  82. B. Gnedenko and A. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley Mathematical Series (Addison-Wesley, Reading, MA, 1954).
  83. J. Nolan, Univariate Stable Distributions: Models for Heavy Tailed Data, Springer Series in Operations Research and Financial Engineering (Springer International Publishing, New York, 2020).
  84. J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep. 195, 127 (1990).
  85. R. Y. Rubinstein, Simulation and the Monte Carlo Method (Wiley, New York, 1981).
  86. A. Alexandru, P. F. Bedaque, A. Carosso, and H. Oh, Infinite variance problem in fermion models, Phys. Rev. D 107, 094502 (2023).
  87. H. Vrcan and J. H. Mentink, Instability of explicit time integration for strongly quenched dynamics with neural quantum states, arXiv:2507.17421.
  88. G. Carleo, K. Choo, D. Hofmann, J. E. Smith, T. Westerhout, F. Alet, E. J. Davis, S. Efthymiou, I. Glasser, S.-H. Lin, M. Mauri, G. Mazzola, C. B. Pereira, and F. Vicentini, netket: A machine learning toolkit for many-body quantum systems, SoftwareX 10, 100311 (2019).
  89. F. Vicentini, D. Hofmann, A. Szabó, D. Wu, C. Roth, C. Giuliani, G. Pescia, J. Nys, V. Vargas-Calderón, N. Astrakhantsev, and G. Carleo, netket3: Machine learning toolbox for many-body quantum systems, SciPost Phys. Codebases 7 (2022).
  90. N. Lambert, E. Giguére, P. Menczel, B. Li, P. Hopf, G. Su’arez, M. Gali, J. Lishman, R. Gadhvi, R. Agarwal, A. Galicia, N. Shammah, P. Nation, J. R. Johansson, S. Ahmed, S. Cross, A. Pitchford, and F. Nori, qutip5: The quantum toolbox in Python, Phys. Rep. 1153, 1 (2026).
  91. Z.-Y. Han, Z.-Q. Wan, and H. Yao, Pfaffian quantum Monte Carlo: Solution to Majorana sign ambiguity and applications, arXiv:2408.10311.
  92. Z.-Q. Wan, X.-D. Dai, and G.-Y. Zhu, Revisiting nishimori multicriticality through the lens of information measures, Phys. Rev. Res. 8, 023059 (2026).
  93. https://github.com/therooler/nqs_blurred_sampling.
  94. D. Ceperley, G. V. Chester, and M. H. Kalos, Monte Carlo simulation of a many-fermion study, Phys. Rev. B 16, 3081 (1977).
  95. L. L. Viteritti, R. Rende, and F. Becca, Transformer variational wave functions for frustrated quantum spin systems, Phys. Rev. Lett. 130, 236401 (2023).
  96. L. L. Viteritti, R. Rende, A. Parola, S. Goldt, and F. Becca, Transformer wave function for two dimensional frustrated magnets: Emergence of a spin-liquid phase in the Shastry-Sutherland model, Phys. Rev. B 111, 134411 (2025).
  97. R. Rende, F. Gerace, A. Laio, and S. Goldt, Mapping of attention mechanisms to a generalized Potts model, Phys. Rev. Res. 6, 023057 (2024).
  98. R. Rende and L. Loris Viteritti, Are queries and keys always relevant? a case study on transformer wave functions, Mach. Learn. 6, 010501 (2025).
  99. R. Wiersema, Jaxmg: A multi-GPU linear solver in jax, arXiv:2601.14466.
  100. M. Solinas, A. Valenti, N. Bou-Rabee, and R. Wiersema, Neural quantum states in mixed precision, arXiv:2601.20782.

Outline

Information

Phys. Rev. X 16, 031059– Published 2 September, 2026
  • Received 8 April 2026
  • Revised 29 June 2026
  • Accepted 10 August 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation