- 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

First- and quasi-second-order optimization algorithms in variational Monte Carlo

Ruojing Peng1,* and Garnet Kin-Lic Chan1,2,†

  • *Contact author: rpeng@fas.harvard.edu
  • †Contact author: gkc1000@gmail.com
PDF

Phys. Rev. Research 7, 043351 – Published 31 December, 2025

DOI: https://doi.org/10.1103/r7c3-zq4b

Abstract

Many quantum many-body wavefunctions, such as Jastrow-Slater, tensor network, and neural quantum states, are studied with the variational Monte Carlo technique, where stochastic optimization is usually performed to obtain a faithful approximation to the ground state of a given Hamiltonian. While first-order gradient descent methods are commonly used for such optimizations, quasi-second-order optimization formulations offer the potential of faster convergence under certain theoretical conditions, but with a similar cost per sample to first-order methods. However, the relative performance of first- and second-order optimizers is influenced in practice by many factors, including the sampling requirements for a faithful optimization step, the influence of wavefunction quality, as well as the wavefunction parametrization and expressivity. Here, we analyze these performance characteristics of first- and quasi-second-order optimization methods for a variety of Hamiltonians, with the additional context of understanding the scaling of these methods (for good performance) as a function of system size. Our findings help clarify the role of first- and quasi-second-order methods in variational Monte Carlo calculations and the conditions under which they should respectively be used. In particular, we find that unlike in deterministic optimization, where closeness to the variational minimum determines the suitability of second-order methods, in stochastic optimization the main factor is the overall expressivity of the wavefunction: Quasi-second-order methods lead to an overall reduction in cost relative to first-order methods when the wavefunction is sufficiently expressive to represent the ground state, even when starting far away from the ground state. This makes quasi-second-order methods an important technique when used with wavefunctions with arbitrarily improvable accuracy.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (99)

  1. W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Quantum Monte Carlo simulations of solids, Rev. Mod. Phys. 73, 33 (2001).
  2. F. Becca and S. Sorella, Quantum Monte Carlo Approaches for Correlated Systems (Cambridge University Press, Cambridge, 2017).
  3. B. M. Austin, D. Y. Zubarev, and W. A. J. Lester, Quantum Monte Carlo and related approaches, Chem. Rev. 112, 263 (2012).
  4. A. Lüchow, Quantum Monte Carlo methods, WIREs Comput. Mol. Sci. 1, 388 (2011).
  5. B. Rubenstein, Introduction to the variational Monte Carlo method in quantum chemistry and physics, in Variational Methods in Molecular Modeling, edited by J. Wu (Springer, Singapore, 2017), pp. 285–313.
  6. G. Bhanot, The metropolis algorithm, Rep. Prog. Phys. 51, 429 (1988).
  7. I. Beichl and F. Sullivan, The metropolis algorithm, Comput. Sci. Eng. 2, 65 (2000).
  8. H. Haario, E. Saksman, and J. Tamminen, An adaptive metropolis algorithm, Bernoulli 7, 223 (2001).
  9. J. E. Gubernatis, Marshall Rosenbluth and the metropolis algorithm, Phys. Plasmas 12, 057303 (2005).
  10. K. L. Mengersen and R. L. Tweedie, Rates of convergence of the Hastings and metropolis algorithms, Ann. Stat. 24, 101 (1996).
  11. M. Casula and S. Sorella, Geminal wavefunctions with Jastrow correlation: A first application to atoms, J. Chem. Phys. 119, 6500 (2003).
  12. B. Tayo and S. Sorella, Exact Jastrow-Slater wave function for the one-dimensional Luttinger model, Phys. Rev. B 78, 115117 (2008).
  13. M. D. Brown, J. R. Trail, P. López Ríos, and R. J. Needs, Energies of the first row atoms from quantum Monte Carlo, J. Chem. Phys. 126, 224110 (2007).
  14. F. Schautz and C. Filippi, Optimized Jastrow–Slater wave functions for ground and excited states: Application to the lowest states of ethene, J. Chem. Phys. 120, 10931 (2004).
  15. P. López Ríos, A. Ma, N. D. Drummond, M. D. Towler, and R. J. Needs, Inhomogeneous backflow transformations in quantum Monte Carlo calculations, Phys. Rev. E 74, 066701 (2006).
  16. X. Lin, H. Zhang, and A. M. Rappe, Optimization of quantum Monte Carlo wave functions using analytical energy derivatives, J. Chem. Phys. 112, 2650 (2000).
  17. C. J. Umrigar and C. Filippi, Energy and variance optimization of many-body wave functions, Phys. Rev. Lett. 94, 150201 (2005).
  18. S. Sorella, Wavefunction optimization in the variational Monte Carlo method, Phys. Rev. B 71, 241103 (2005).
  19. C. J. Umrigar, J. Toulouse, C. Filippi, S. Sorella, and R. G. Hennig, Alleviation of the fermion-sign problem by optimization of many-body wave functions, Phys. Rev. Lett. 98, 110201 (2007).
  20. J. Toulouse and C. J. Umrigar, Optimization of quantum Monte Carlo wave functions by energy minimization, J. Chem. Phys. 126, (2007).
  21. J. Toulouse and C. J. Umrigar, Full optimization of Jastrow–Slater wave functions with application to the first-row atoms and homonuclear diatomic molecules, J. Chem. Phys. 128, 174101 (2008).
  22. I. Sabzevari, A. Mahajan, and S. Sharma, An accelerated linear method for optimizing non-linear wavefunctions in variational Monte Carlo, J. Chem. Phys. 152, 024111 (2020).
  23. A. W. Sandvik and G. Vidal, Variational quantum Monte Carlo simulations with tensor-network states, Phys. Rev. Lett. 99, 220602 (2007).
  24. N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Simulation of quantum many-body systems with strings of operators and Monte Carlo tensor contractions, Phys. Rev. Lett. 100, 040501 (2008).
  25. L. Wang, I. Pižorn, and F. Verstraete, Monte Carlo simulation with tensor network states, Phys. Rev. B 83, 134421 (2011).
  26. W.-Y. Liu, S.-J. Dong, Y.-J. Han, G.-C. Guo, and L. He, Gradient optimization of finite projected entangled pair states, Phys. Rev. B 95, 195154 (2017).
  27. 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).
  28. Z. Y. Xie, J. Chen, J. F. Yu, X. Kong, B. Normand, and T. Xiang, Tensor renormalization of quantum many-body systems using projected entangled simplex states, Phys. Rev. X 4, 011025 (2014).
  29. R. Orús and G. Vidal, Simulation of two-dimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction, Phys. Rev. B 80, 094403 (2009).
  30. R. Orús, Tensor networks for complex quantum systems, Nat. Rev. Phys. 1, 538 (2019).
  31. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117 (2014).
  32. N. Schuch, I. Cirac, and D. Pérez-García, Peps as ground states: Degeneracy and topology, Ann. Phys. 325, 2153 (2010).
  33. M. Levin and C. P. Nave, Tensor renormalization group approach to two-dimensional classical lattice models, Phys. Rev. Lett. 99, 120601 (2007).
  34. Z.-C. Gu, M. Levin, and X.-G. Wen, Tensor-entanglement renormalization group approach as a unified method for symmetry breaking and topological phase transitions, Phys. Rev. B 78, 205116 (2008).
  35. Z.-C. Gu and X.-G. Wen, Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order, Phys. Rev. B 80, 155131 (2009).
  36. H. C. Jiang, Z. Y. Weng, and T. Xiang, Accurate determination of tensor network state of quantum lattice models in two dimensions, Phys. Rev. Lett. 101, 090603 (2008).
  37. A. Sfondrini, J. Cerrillo, N. Schuch, and J. I. Cirac, Simulating two- and three-dimensional frustrated quantum systems with string-bond states, Phys. Rev. B 81, 214426 (2010).
  38. I. Glasser, N. Pancotti, M. August, I. D. Rodriguez, and J. I. Cirac, Neural-network quantum states, string-bond states, and chiral topological states, Phys. Rev. X 8, 011006 (2018).
  39. J. Thibaut, T. Roscilde, and F. Mezzacapo, Long-range entangled-plaquette states for critical and frustrated quantum systems on a lattice, Phys. Rev. B 100, 155148 (2019).
  40. F. Mezzacapo, N. Schuch, M. Boninsegni, and J. I. Cirac, Ground-state properties of quantum many-body systems: Entangled-plaquette states and variational Monte Carlo, New J. Phys. 11, 083026 (2009).
  41. J. Carrasquilla, G. Chen, and R. G. Melko, Tripartite entangled plaquette state in a cluster magnet, Phys. Rev. B 96, 054405 (2017).
  42. F. Mezzacapo, A. Angelone, and G. Pupillo, Two holes in a two-dimensional quantum antiferromagnet: A variational study based on entangled-plaquette states, Phys. Rev. B 94, 155120 (2016).
  43. F. Mezzacapo and J. I. Cirac, Ground-state properties of the spin- antiferromagnetic Heisenberg model on the triangular lattice: A variational study based on entangled-plaquette states, New J. Phys. 12, 103039 (2010).
  44. W.-Y. Liu, S.-J. Du, R. Peng, J. Gray, and G. K.-L. Chan, Tensor network computations that capture strict variationality, volume law behavior, and the efficient representation of neural network states, Phys. Rev. Lett. 133, 260404 (2024).
  45. A. Chen and M. Heyl, Empowering deep neural quantum states through efficient optimization, Nat. Phys. 20, 1476 (2024).
  46. R. Rende, L. L. Viteritti, L. Bardone, F. Becca, and S. Goldt, A simple linear algebra identity to optimize large-scale neural network quantum states, Commun. Phys. 7, 260 (2024).
  47. R. J. Webber and M. Lindsey, Rayleigh-Gauss-Newton optimization with enhanced sampling for variational Monte Carlo, Phys. Rev. Res. 4, 033099 (2022).
  48. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  49. O. Sharir, A. Shashua, and G. Carleo, Neural tensor contractions and the expressive power of deep neural quantum states, Phys. Rev. B 106, 205136 (2022).
  50. D.-L. Deng, X. Li, and S. Das Sarma, Quantum entanglement in neural network states, Phys. Rev. X 7, 021021 (2017).
  51. K. Choo, A. Mezzacapo, and G. Carleo, Fermionic neural-network states for ab-initio electronic structure, Nat. Commun. 11, 2368 (2020).
  52. T. Zhao, J. Stokes, and S. Veerapaneni, Scalable neural quantum states architecture for quantum chemistry, Mach. Learn.: Sci. Technol. 4, 025034 (2023).
  53. Z.-A. Jia, B. Yi, R. Zhai, Y.-C. Wu, G.-C. Guo, and G.-P. Guo, Quantum neural network states: A brief review of methods and applications, Adv. Quantum Technol. 2, 1800077 (2019).
  54. M. Reh, M. Schmitt, and M. Gärttner, Optimizing design choices for neural quantum states, Phys. Rev. B 107, 195115 (2023).
  55. H.-K. Zhang, C. Zhu, M. Jing, and X. Wang, Statistical analysis of quantum state learning process in quantum neural networks, in Advances in Neural Information Processing Systems, edited by A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine (Curran Associates, Inc., Red Hook, NY, USA, 2023), Vol. 36, pp. 33133–33160.
  56. J. Duchi, E. Hazan, and Y. Singer, Adaptive subgradient methods for online learning and stochastic optimization, J. Mach. Learn. Res. 12, 2121 (2011).
  57. H. B. McMahan and M. J. Streeter, Adaptive bound optimization for online convex optimization, in Proceedings of the 23rd Annual Conference On Learning Theory (Omnipress, Madison, WI, USA, 2010).
  58. D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, in Proceedings of 3rd International Conference on Learning Representations (OpenReview.net, Amherst, MA, USA, 2015).
  59. M. D. Zeiler, ADADELTA: An adaptive learning rate method, arXiv:1212.5701.
  60. T. Dozat, Incorporating Nesterov momentum into Adam, in Proceedings of the 4th International Conference on Learning Representations (OpenReview.net, Amherst, MA, USA, 2016), pp. 1–4.
  61. S. J. Reddi, S. Kale, and S. Kumar, On the convergence of Adam and beyond, in 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, Conference Track Proceedings (OpenReview.net, Amherst, MA, USA, 2018).
  62. I. Sabzevari and S. Sharma, Improved speed and scaling in orbital space variational Monte Carlo, J. Chem. Theory Comput. 14, 6276 (2018).
  63. S. Sorella, Green function Monte Carlo with stochastic reconfiguration, Phys. Rev. Lett. 80, 4558 (1998).
  64. S. Sorella, Generalized lanczos algorithm for variational quantum Monte Carlo, Phys. Rev. B 64, 024512 (2001).
  65. M. Casula, C. Attaccalite, and S. Sorella, Correlated geminal wave function for molecules: An efficient resonating valence bond approach, J. Chem. Phys. 121, 7110 (2004).
  66. E. Neuscamman, C. J. Umrigar, and G. K.-L. Chan, Optimizing large parameter sets in variational quantum Monte Carlo, Phys. Rev. B 85, 045103 (2012).
  67. J. Stokes, J. Izaac, N. Killoran, and G. Carleo, Quantum natural gradient, Quantum 4, 269 (2020).
  68. J. Nocedal and S. J. Wright, Numerical Optimization (Springer, New York, NY, 2006).
  69. L. Zhao and E. Neuscamman, A blocked linear method for optimizing large parameter sets in variational Monte Carlo, J. Chem. Theory Comput. 13, 2604 (2017).
  70. F. Dangel, J. Müller, and M. Zeinhofer, Kronecker-factored approximate curvature for physics-informed neural networks, in Proceedings of the Thirty-eighth Annual Conference on Neural Information Processing Systems (OpenReview.net, Amherst, MA, USA, 2024).
  71. J. Martens and R. Grosse, Optimizing neural networks with Kronecker-factored approximate curvature, in Proceedings of the 32nd International Conference on International Conference on Machine Learning, ICML'15 Vol. 37 (JMLR.org, Lille, France, 2015).
  72. A. Botev and J. Martens, KFAC-JAX (2022), https://github.com/google-deepmind/kfac-jax.
  73. M. W. Lee, M. Mella, and A. M. Rappe, Electronic quantum Monte Carlo calculations of atomic forces, vibrations, and anharmonicities, J. Chem. Phys. 122, 244103 (2005).
  74. C.-Y. Park and M. J. Kastoryano, Geometry of learning neural quantum states, Phys. Rev. Res. 2, 023232 (2020).
  75. C. J. Umrigar, Observations on variational and projector Monte Carlo methods, J. Chem. Phys. 143, 164105 (2015).
  76. D. Ceperley, G. V. Chester, and M. H. Kalos, Monte Carlo simulation of a many-fermion study, Phys. Rev. B 16, 3081 (1977).
  77. J. Kolorenč and L. Mitas, Applications of quantum Monte Carlo methods in condensed systems, Rep. Prog. Phys. 74, 026502 (2011).
  78. M. Nightingale and C. Umrigar, Quantum Monte Carlo Methods in Physics and Chemistry, NATO ASI Series: Mathematical and Physical Sciences (Springer, Netherlands, 1998).
  79. B. L. Hammond, W. A. Lester, and P. J. Reynolds, Monte Carlo Methods in ab initio Quantum Chemistry (World Scientific, Singapore, 1994).
  80. J. Gubernatis, N. Kawashima, and P. Werner, Quantum Monte Carlo Methods: Algorithms for Lattice Models (Cambridge University Press, Cambridge, 2016).
  81. M. Bukov, M. Schmitt, and M. Dupont, Learning the ground state of a non-stoquastic quantum Hamiltonian in a rugged neural network landscape, SciPost Phys. 10, 147 (2021).
  82. F. Verstraete, V. Murg, and J. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57, 143 (2008).
  83. M. Lubasch, J. I. Cirac, and M.-C. Bañuls, Algorithms for finite projected entangled pair states, Phys. Rev. B 90, 064425 (2014).
  84. C. C. Paige and M. A. Saunders, Solution of sparse indefinite systems of linear equations, SIAM J. Numer. Anal. 12, 617 (1975).
  85. S.-C. T. Choi, C. C. Paige, and M. A. Saunders, MINRES-QLP: A Krylov subspace method for indefinite or singular symmetric systems, SIAM J. Sci. Comput. 33, 1810 (2011).
  86. Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7, 856 (1986).
  87. A. H. Baker, E. R. Jessup, and T. Manteuffel, A technique for accelerating the convergence of restarted GMRES, SIAM J. Matrix Anal. Appl. 26, 962 (2005).
  88. M. P. Nightingale and V. Melik-Alaverdian, Optimization of ground- and excited-state wave functions and van der Waals clusters, Phys. Rev. Lett. 87, 043401 (2001).
  89. E. Neuscamman, Improved optimization for the cluster Jastrow antisymmetric geminal power and tests on triple-bond dissociations, J. Chem. Theory Comput. 12, 3149 (2016).
  90. B. Polyak, Introduction to Optimization (Optimization Software, Inc., New York, NY, USA, 1987).
  91. C. K. Majumdar and D. K. Ghosh, On next‐nearest‐neighbor interaction in linear chain. I, J. Math. Phys. 10, 1388 (1969).
  92. C. K. Majumdar and D. K. Ghosh, On next‐nearest‐neighbor interaction in linear chain. II, J. Math. Phys. 10, 1399 (1969).
  93. A. Lavarélo and G. Roux, Spinon excitation spectra of the J1−J2 chain from analytical calculations in the dimer basis and exact diagonalization, Eur. Phys. J. B 87, 229 (2014).
  94. N. Chancellor and S. Haas, Propagation of disturbances in degenerate quantum systems, Phys. Rev. B 84, 035130 (2011).
  95. F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Criticality, the area law, and the computational power of projected entangled pair states, Phys. Rev. Lett. 96, 220601 (2006).
  96. H. Zhai, H. R. Larsson, S. Lee, Z.-H. Cui, T. Zhu, C. Sun, L. Peng, R. Peng, K. Liao, J. Tölle, J. Yang, S. Li, and G. K.-L. Chan, Block2: A comprehensive open source framework to develop and apply state-of-the-art DMRG algorithms in electronic structure and beyond, J. Chem. Phys. 159, 234801 (2023).
  97. W.-Y. Liu, S. Dong, C. Wang, Y. Han, H. An, G.-C. Guo, and L. He, Gapless spin liquid ground state of the spin-12J1−J2 Heisenberg model on square lattices, Phys. Rev. B 98, 241109 (2018).
  98. W.-J. Hu, F. Becca, A. Parola, and S. Sorella, Direct evidence for a gapless Z2 spin liquid by frustrating Néel antiferromagnetism, Phys. Rev. B 88, 060402 (2013).
  99. R. Peng and G. K.-L. Chan, Dataset for “An analysis of first- and quasi-second-order optimization algorithms in variational Monte Carlo (2025)”, https://github.com/r-peng/quimb_vmc.

Outline

Information

Phys. Rev. Research 7, 043351– Published 31 December, 2025
  • Received 20 August 2025
  • Accepted 7 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