- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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

Improved energies and local energies with weighted variational Monte Carlo

Huan Zhang1, Robert J. Webber2, Michael Lindsey3, Timothy C. Berkelbach4,5,*, and Jonathan Weare1,†

  • *Contact author: t.berkelbach@columbia.edu
  • †Contact author: weare@nyu.edu
PDF

Phys. Rev. Research 8, 013213 – Published 25 February, 2026

DOI: https://doi.org/10.1103/6b3v-38jp

Abstract

Neural network parametrizations have increasingly been used to represent the ground and excited states in variational Monte Carlo (VMC) with promising results. However, traditional VMC methods only optimize the wave function in regions of peak probability. The wave function is uncontrolled in the tails of the probability distribution, which can limit the accuracy of the trained wave function. To improve the approximation accuracy in the probability tails, this paper interprets VMC as a gradient flow in the space of wave functions, followed by a projection step. From this perspective, arbitrary probability distributions can be used in the projection step, allowing the user to prioritize accuracy in different regions of state space. Motivated by this theoretical perspective, the paper tests a weighted VMC method on the antiferromagnetic Heisenberg model for a periodic spin chain. Compared to traditional VMC, weighted VMC reduces the error in the ground state energy by a factor of 2, and it reduces the errors in the local energies away from the mode by large factors of 102–104.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (39)

  1. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  2. D. Luo and B. K. Clark, Backflow transformations via neural networks for quantum many-body wave functions, Phys. Rev. Lett. 122, 226401 (2019).
  3. 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).
  4. J. Hermann, Z. Schätzle, and F. Noé, Deep-neural-network solution of the electronic Schrödinger equation, Nat. Chem. 12, 891 (2020).
  5. J. Gubernatis, N. Kawashima, and P. Werner, Quantum Monte Carlo Methods: Algorithms for Lattice Models (Cambridge University Press, Cambridge, UK, 2016).
  6. F. Becca and S. Sorella, Quantum Monte Carlo Approaches for Correlated Systems (Cambridge University Press, Cambridge, UK, 2017).
  7. 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).
  8. Y. Nomura, Helping restricted Boltzmann machines with quantum-state representation by restoring symmetry, J. Phys.: Condens. Matter 33, 174003 (2021).
  9. 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).
  10. 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).
  11. C. Filippi, M. Zaccheddu, and F. Buda, Absorption spectrum of the green fluorescent protein chromophore: A difficult case for ab initio methods? J. Chem. Theory Comput. 5, 2074 (2009).
  12. P. M. Zimmerman, J. Toulouse, Z. Zhang, C. B. Musgrave, and C. J. Umrigar, Excited states of methylene from quantum Monte Carlo, J. Chem. Phys. 131, 124103 (2009).
  13. L. Zhao and E. Neuscamman, An efficient variational principle for the direct optimization of excited states, J. Chem. Theory Comput. 12, 3436 (2016).
  14. T. Westerhout, N. Astrakhantsev, K. S Tikhonov, M. I Katsnelson, and A. A Bagrov, Generalization properties of neural network approximations to frustrated magnet ground states, Nat. Commun. 11, 1593 (2020).
  15. R. J. Webber and M. Lindsey, Rayleigh-Gauss-Newton optimization with enhanced sampling for variational Monte Carlo, Phys. Rev. Res. 4, 033099 (2022).
  16. H. Zhang, R. J. Webber, M. Lindsey, T. C. Berkelbach, and J. Weare, Understanding and eliminating spurious modes in variational Monte Carlo using collective variables, Phys. Rev. Res. 5, 023101 (2023).
  17. C.-Y. Park and M. J. Kastoryano, Geometry of learning neural quantum states, Phys. Rev. Res. 2, 023232 (2020).
  18. M. Karbach, K. Hu, and G. Müller, Introduction to the Bethe ansatz, II, Comput. Phys. 12, 565 (1998).
  19. J. Stokes, J. Izaac, N. Killoran, and G. Carleo, Quantum natural gradient, Quantum 4, 269 (2020).
  20. K. Neklyudov, J. Nys, L. Thiede, J. Carrasquilla, Q. Liu, M. Welling, and A. Makhzani, Wasserstein quantum Monte Carlo: A novel approach for solving the quantum many-body Schrödinger equation, in Advances in Neural Information Processing Systems (Curran Associates, Inc., New York, 2023), Vol. 36, pp. 63461–63482.
  21. P. A. M. Dirac, Note on exchange phenomena in the Thomas atom, Math. Proc. Cambridge Philos. Soc. 26, 376 (1930).
  22. J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete, Unifying time evolution and optimization with matrix product states, Phys. Rev. B 94, 165116 (2016).
  23. A. Laio and M. Parrinello, Escaping free-energy minima, Proc. Natl. Acad. Sci. USA 99, 12562 (2002).
  24. A. Barducci, G. Bussi, and M. Parrinello, Well-tempered metadynamics: A smoothly converging and tunable free-energy method, Phys. Rev. Lett. 100, 020603 (2008).
  25. R. F. Bishop and D. J. J. Farnell, Marshall-Peierls sign rules, the quantum Monte Carlo method, and frustration, in Recent Progress in Many-Body Theories (World Scientific, Singapore, 2000), pp. 457–460.
  26. T. Tieleman, Training restricted Boltzmann machines using approximations to the likelihood gradient, in Proceedings of the 25th International Conference on Machine Learning, ICML’08 (Association for Computing Machinery, New York, NY, USA, 2008), pp. 1064–1071.
  27. K. Sohn and H. Lee, Learning invariant representations with local transformations, in Proceedings of the 29th International Conference on Machine Learning, ICML’12 (Omnipress, Madison, WI, USA, 2012), pp. 1339–1346.
  28. M. Norouzi, M. Ranjbar, and G. Mori, Stacks of convolutional restricted Boltzmann machines for shift-invariant feature learning, in 2009 IEEE Conference on Computer Vision and Pattern Recognition (IEEE, Miami, FL, USA, 2009), pp. 2735–2742.
  29. S. Sorella, Generalized Lanczos algorithm for variational quantum Monte Carlo, Phys. Rev. B 64, 024512 (2001).
  30. S. Sorella, M. Casula, and D. Rocca, Weak binding between two aromatic rings: Feeling the van der Waals attraction by quantum Monte Carlo methods, J. Chem. Phys. 127, 014105 (2007).
  31. W. Wirtinger, Zur formalen Theorie der Funktionen von mehr komplexen Veränderlichen, Math. Ann. 97, 357 (1927).
  32. J. Bruna, B. Peherstorfer, and E. Vanden-Eijnden, Neural Galerkin schemes with active learning for high-dimensional evolution equations, J. Comput. Phys. 496, 112588 (2024).
  33. H. Zhang, Y. Chen, E. Vanden-Eijnden, and B. Peherstorfer, Sequential-in-time training of nonlinear parametrizations for solving time-dependent partial differential equations, SIAM review (to be published).
  34. J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, JAX composable transformations of Python+NumPy programs (2018), http://github.com/google/jax.
  35. Huan Zhang, Weighted VMC, GitHub, 2025, https://github.com/hzhang-math/WeightedVMC.
  36. J. Martens and R. Grosse, Optimizing neural networks with Kronecker-factored approximate curvature, in Proceedings of the 32nd International Conference on Machine Learning, edited by F. Bach and D. Blei, in Proceedings of Machine Learning Research (PMLR, Lille, France, 2015), Vol. 37, pp. 2408–2417.
  37. A. Chen and M. Heyl, Empowering deep neural quantum states through efficient optimization, Nat. Phys. 20, 1476 (2024).
  38. G. Goldshlager, N. Abrahamsen, and L. Lin, A Kaczmarz-inspired approach to accelerate the optimization of neural network wavefunctions, J. Comput. Phys. 516, 113351 (2024).
  39. https://github.com/hzhang-math/WeightedVMC.

Outline

Information

Phys. Rev. Research 8, 013213– Published 25 February, 2026
  • Received 16 June 2025
  • Accepted 17 December 2025
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