- Physical Review Journals
All JournalsPhysics Magazine

PRX Intelligence

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

Pareto Frontier of Neural Quantum States: Scalable, Affordable, and Accurate Convolutional Backflow for Strongly Correlated Lattice Fermions

Yuntian Gu1,2,*, Zeyao Han2,3,*, Wenrui Li1,2,*, Zhiyu Xiao4, Tao Xiang4,†, Mingpu Qin5,‡, Liwei Wang1,§, and Dingshun Lv2,∥

  • *These authors contributed equally to this work.
  • †Contact author: txiang@iphy.ac.cn
  • ‡Contact author: qinmingpu@sjtu.edu.cn
  • §Contact author: wanglw@pku.edu.cn
  • ∥Contact author: ywlds163@gmail.com
PDF

PRX Intelligence 1, 023001 – Published 6 October, 2026

DOI: https://doi.org/10.1103/2kq2-5g5q

Abstract

The recent development of Neural Quantum States (NQS) has established them as one of the most accurate methods for studying strongly correlated many-fermion systems, outperforming existing many-body approaches for large systems. However, NQS calculations are currently highly resource intensive. In this work, we introduce an improved Pareto frontier of efficiency and accuracy for NQS in the simulation of strongly correlated lattice fermions. This frontier is defined by two complementary backflow-related architectures: the Sparse Convolutional Ansatz for Lattice Electrons (SCALE), which achieves high computational efficiency, and the Accurate Convolutional ansatz for lattice Electrons (ACE), which delivers high variational accuracy, based on benchmark results on the Hubbard model and the t–J model on large lattice sizes. SCALE utilizes a tailored convolutional design to enable an efficient local update based on the low-rank update of determinants. This structure reduces the computational scaling from O(N4) to O(N3) in backflow methods, resulting in a practical speedup exceeding 40× in our test cases while retaining accurate variational accuracy. As an application, we study the 1/8-doped pure Hubbard model up to size 32×32 and find no significant energy difference between the horizontal and vertical unit-filled stripe states, in contrast to the half-filled stripe state when next-nearest-neighbor hoppings are included. ACE, conversely, employs a deep convolutional stack to maximize expressive power, achieving high accuracy on large systems. We perform extensive benchmark calculations on the Hubbard and t–J models with the two architectures. On the challenging repulsive Hubbard model, SCALE achieves variational energies competitive with leading methods at a fraction of the computational cost. Meanwhile, ACE achieves lower variational energies than previously reported results while requiring only one-sixth of the run-time for system size 16×4. These NQS approaches provide scalable, affordable, and accurate tools for exploring the rich physics of strongly correlated fermionic systems.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (50)

  1. E. H. Lieb and F. Y. Wu, Absence of Mott transition in an exact solution of the short-range, one-band model in one dimension, Phys. Rev. Lett. 20, 1445 (1968).
  2. E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Colloquium: Theory of intertwined orders in high temperature superconductors, Rev. Mod. Phys. 87, 457 (2015).
  3. M. Qin, T. Schäfer, S. Andergassen, P. Corboz, and E. Gull, The Hubbard model: A computational perspective, Annu. Rev. Condens. Matter Phys. 13, 275 (2022).
  4. D. P. Arovas, E. Berg, S. A. Kivelson, and S. Raghu, The Hubbard model, Annu. Rev. Condens. Matter Phys. 13, 239 (2022).
  5. C.-M. Chung, M. Qin, S. Zhang, U. Schollwöck, and S. R. White (The Simons Collaboration on the Many-Electron Problem), Plaquette versus ordinary d-wave pairing in the t′-Hubbard model on a width-4 cylinder, Phys. Rev. B 102, 041106 (2020).
  6. H. Xu, C.-M. Chung, M. Qin, U. Schollwöck, S. R. White, and S. Zhang, Coexistence of superconductivity with partially filled stripes in the Hubbard model, Science 384, eadh7691 (2024).
  7. Y. Shen, X. Qian, and M. Qin, Ground state of electron-doped t-t′-j model on cylinders: An investigation of finite size and boundary condition effects, Chin. Phys. B 34, 087105 (2025).
  8. X. Lu, F. Chen, W. Zhu, D. N. Sheng, and S.-S. Gong, Emergent superconductivity and competing charge orders in hole-doped square-lattice t–j model, Phys. Rev. Lett. 132, 066002 (2024).
  9. H.-C. Jiang, T. P. Devereaux, and S. A. Kivelson, Competition between charge-density-wave and superconducting orders on eight-leg square Hubbard cylinders, arXiv:2511.18644.
  10. J. P. LeBlanc, A. E. Antipov, F. Becca, I. W. Bulik, G. K.-L. Chan, C.-M. Chung, Y. Deng, M. Ferrero, T. M. Henderson, C. A. Jiménez-Hoyos, et al., Solutions of the two-dimensional Hubbard model: Benchmarks and results from a wide range of numerical algorithms, Phys. Rev. X 5, 041041 (2015).
  11. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  12. S. Östlund and S. Rommer, Thermodynamic limit of density matrix renormalization, Phys. Rev. Lett. 75, 3537 (1995).
  13. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  14. J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021).
  15. F. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, arXiv:cond-mat/0407066.
  16. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117 (2014).
  17. T. Xiang, Density Matrix and Tensor Network Renormalization (Cambridge University Press, Cambridge, England, 2023).
  18. G. Carleo and M. Troyer, Solving the quantum many-body problem with artificial neural networks, Science 355, 602 (2017).
  19. D. Luo and B. K. Clark, Backflow transformations via neural networks for quantum many-body wave functions, Phys. Rev. Lett. 122, 226401 (2019).
  20. J. Robledo Moreno, G. Carleo, A. Georges, and J. Stokes, Fermionic wave functions from neural-network constrained hidden states, Proc. Natl. Acad. Sci. USA 119, e2122059119 (2022).
  21. 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, Nat. Commun. 17, 7838 (2026).
  22. X. Liang, Investigating the Fermi-Hubbard model by the tensor-backflow method, Phys. Rev. B 113, 155121 (2026).
  23. 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.
  24. C. Roth, A. Chen, A. Sengupta, and A. Georges, Superconductivity in the two-dimensional Hubbard model revealed by neural quantum states, arXiv:2511.07566.
  25. K. Loehr and B. K. Clark, Enhancing neural network backflow, arXiv:2510.26906.
  26. R. Blankenbecler, D. J. Scalapino, and R. L. Sugar, Monte Carlo calculations of coupled boson-fermion systems. I, Phys. Rev. D 24, 2278 (1981).
  27. J. E. Hirsch, Two-dimensional Hubbard model: Numerical simulation study, Phys. Rev. B 31, 4403 (1985).
  28. J. E. Hirsch, Discrete Hubbard-Stratonovich transformation for fermion lattice models, Phys. Rev. B 28, 4059 (1983).
  29. Y.-Y. He, H. Shi, and S. Zhang, Reaching the continuum limit in finite-temperature ab initio field-theory computations in many-fermion systems, Phys. Rev. Lett. 123, 136402 (2019).
  30. 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).
  31. M. Scherbela, N. Gao, P. Grohs, and S. Günnemann, Accurate ab-initio neural-network solutions to large-scale electronic structure problems, arXiv:2504.06087.
  32. E. Wigner, On the interaction of electrons in metals, Phys. Rev. 46, 1002 (1934).
  33. R. P. Feynman and M. Cohen, Energy spectrum of the excitations in liquid helium, Phys. Rev. 102, 1189 (1956).
  34. L. F. Tocchio, F. Becca, A. Parola, and S. Sorella, Role of backflow correlations for the nonmagnetic phase of the t–t′ Hubbard model, Phys. Rev. B 78, 041101 (2008).
  35. K. He, X. Zhang, S. Ren, and J. Sun, Deep residual learning for image recognition, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, Piscataway, NJ, 2016), pp. 770–778.
  36. S. Elfwing, E. Uchibe, and K. Doya, Sigmoid-weighted linear units for neural network function approximation in reinforcement learning, Neural Netw. 107, 3 (2018).
  37. S. R. White, D. J. Scalapino, R. L. Sugar, E. Y. Loh, J. E. Gubernatis, and R. T. Scalettar, Numerical study of the two-dimensional Hubbard model, Phys. Rev. B 40, 506 (1989).
  38. J. L. Ba, J. R. Kiros, and G. E. Hinton, Layer normalization, arXiv:1607.06450.
  39. A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly, et al., An image is worth 16 × 16 words: Transformers for image recognition at scale, in Proceedings of the International Conference on Learning Representations (ICLR) (ICLR, 2021).
  40. N. Trivedi and D. M. Ceperley, Ground-state correlations of quantum antiferromagnets: A Green-function Monte Carlo study, Phys. Rev. B 41, 4552 (1990).
  41. D. F. B. ten Haaf, H. J. M. van Bemmel, J. M. J. van Leeuwen, W. van Saarloos, and D. M. Ceperley, Proof for an upper bound in fixed-node Monte Carlo for lattice fermions, Phys. Rev. B 51, 13039 (1995).
  42. J. Hubbard, Electron correlations in narrow energy bands, Proc. R. Soc. London A 276, 238 (1963).
  43. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Microscopic theory of superconductivity, Phys. Rev. 106, 162 (1957).
  44. R. T. Scalettar, E. Y. Loh, J. E. Gubernatis, A. Moreo, S. R. White, D. J. Scalapino, R. L. Sugar, and E. Dagotto, Phase diagram of the two-dimensional negative-u Hubbard model, Phys. Rev. Lett. 62, 1407 (1989).
  45. C. N. Yang, Concept of off-diagonal long-range order and the quantum phases of liquid He and of superconductors, Rev. Mod. Phys. 34, 694 (1962).
  46. F. Zhang and T. Rice, Effective Hamiltonian for the superconducting Cu oxides, Phys. Rev. B 37, 3759 (1988).
  47. L. Sharma, A. Shokry, R. Nutakki, O. Simard, M. Ferrero, and F. Vicentini, Comparing symmetrized determinant neural quantum states for the Hubbard model, Phys. Rev. B 113, 245104 (2026).
  48. ByteDance, LAQX, Github, 2026, https://github.com/bytedance/laqx.
  49. Y. Gu, Data for “Pareto frontier of neural quantum states: Scalable, affordable, and accurate convolutional backflow for strongly correlated lattice fermions”, Github, 2026, https://github.com/guyuntian/open_data.
  50. 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).

Outline

Information

PRX Intelligence 1, 023001– Published 6 October, 2026
  • Received 10 February 2026
  • Accepted 11 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