Accurate neural quantum states for interacting lattice bosons

Zakari Denis; Giuseppe Carleo

doi:10.22331/q-2025-06-17-1772

Accurate neural quantum states for interacting lattice bosons

Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
Center for Quantum Science and Engineering, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

Published:	2025-06-17, volume 9, page 1772
Editor:	Ángela Capel
Eprint:	arXiv:2404.07869v2
Doi:	https://doi.org/10.22331/q-2025-06-17-1772
Citation:	Quantum 9, 1772 (2025).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

In recent years, neural quantum states have emerged as a powerful variational approach, achieving state-of-the-art accuracy when representing the ground-state wave function of a great variety of quantum many-body systems, including spin lattices, interacting fermions or continuous-variable systems. However, accurate neural representations of the ground state of interacting bosons on a lattice have remained elusive. We introduce a neural backflow Jastrow Ansatz, in which occupation factors are dressed with translationally equivariant many-body features generated by a deep neural network. We show that this neural quantum state is able to faithfully represent the ground state of the 2D Bose-Hubbard Hamiltonian across all values of the interaction strength. We scale our simulations to lattices of dimension up to $20{\times}20$ while achieving the best variational energies reported for this model. This enables us to investigate the scaling of the entanglement entropy across the superfluid-to-Mott quantum phase transition, a quantity hard to extract with non-variational approaches.

Featured image: The deep neural backflow-Jastrow Ansatz efficiently captures complex quantum correlations by expressing them as effective two-body interactions between particle occupations dressed with learned many-body translationally equivariant features.

Popular summary

Understanding the complex behavior of interacting bosons, like ultracold atoms hopping on an optical lattice, has long posed a challenge for both numerical physicists. In our work, we introduce a new type of neural quantum state that overcomes this difficulty.

By combining physical insight and the expressive power of deep neural networks, we develop a neural backflow-Jastrow Ansatz. This approach allows the network to learn how particles influence each other through many-body correlations, while respecting the symmetries of the system. Using this method, we accurately capture the ground state of the two-dimensional Bose-Hubbard model across all interaction strengths.

We scale our simulations lattices as large as $20{\times}20$ sites and achieve the best variational energies reported to date for this system. This high level of accuracy also let us probe subtle quantum phenomena, such as how entanglement behaves at the superfluid-to- Mott insulator phase transition, revealing the signature of an underlying Higgs mode. This had previously proved challenging to uncover directly with traditional computational techniques.

► BibTeX data

@article{Denis2025accurateneural,
  doi = {10.22331/q-2025-06-17-1772},
  url = {https://doi.org/10.22331/q-2025-06-17-1772},
  title = {Accurate neural quantum states for interacting lattice bosons},
  author = {Denis, Zakari and Carleo, Giuseppe},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1772},
  month = jun,
  year = {2025}
}

► References

[1] Ghassan George Batrouni, Richard T. Scalettar, and Gergely T. Zimanyi. Quantum critical phenomena in one-dimensional Bose systems. Physical Review Letters, 65 (14): 1765–1768, 1990. 10.1103/PhysRevLett.65.1765.
https://doi.org/10.1103/PhysRevLett.65.1765

[2] Werner Krauth, Nandini Trivedi, and David Ceperley. Superfluid-insulator transition in disordered boson systems. Physical Review Letters, 67 (17): 2307–2310, 1991. 10.1103/PhysRevLett.67.2307.
https://doi.org/10.1103/PhysRevLett.67.2307

[3] Richard T. Scalettar, Ghassan George Batrouni, and Gergely T. Zimanyi. Localization in interacting, disordered, Bose systems. Physical Review Letters, 66 (24): 3144–3147, 1991. 10.1103/PhysRevLett.66.3144.
https://doi.org/10.1103/PhysRevLett.66.3144

[4] Ghassan George Batrouni and Richard T. Scalettar. World-line quantum Monte Carlo algorithm for a one-dimensional Bose model. Physical Review B, 46 (14): 9051–9062, 1992. 10.1103/PhysRevB.46.9051.
https://doi.org/10.1103/PhysRevB.46.9051

[5] B. Capogrosso-Sansone, N. V. Prokof'ev, and B. V. Svistunov. Phase diagram and thermodynamics of the three-dimensional Bose-Hubbard model. Physical Review B, 75 (13): 134302, 2007. 10.1103/PhysRevB.75.134302.
https://doi.org/10.1103/PhysRevB.75.134302

[6] Barbara Capogrosso-Sansone, Şebnem Güneş Söyler, Nikolay Prokof'ev, and Boris Svistunov. Monte Carlo study of the two-dimensional Bose-Hubbard model. Physical Review A, 77 (1): 015602, 2008. 10.1103/PhysRevA.77.015602.
https://doi.org/10.1103/PhysRevA.77.015602

[7] Nandini Trivedi and D. M. Ceperley. Ground-state correlations of quantum antiferromagnets: A Green-function Monte Carlo study. Physical Review B, 41 (7): 4552–4569, 1990. 10.1103/PhysRevB.41.4552.
https://doi.org/10.1103/PhysRevB.41.4552

[8] Matteo Calandra Buonaura and Sandro Sorella. Numerical study of the two-dimensional Heisenberg model using a Green function Monte Carlo technique with a fixed number of walkers. Physical Review B, 57 (18): 11446–11456, 1998. 10.1103/PhysRevB.57.11446.
https://doi.org/10.1103/PhysRevB.57.11446

[9] Federico Becca and Sandro Sorella. Quantum Monte Carlo Approaches for Correlated Systems. Cambridge University Press, 1st edition, 2017. 10.1017/9781316417041.
https://doi.org/10.1017/9781316417041

[10] Stefano Baroni and Saverio Moroni. Reptation Quantum Monte Carlo: A Method for Unbiased Ground-State Averages and Imaginary-Time Correlations. Physical Review Letters, 82 (24): 4745–4748, 1999. 10.1103/PhysRevLett.82.4745.
https://doi.org/10.1103/PhysRevLett.82.4745

[11] Giuseppe Carleo, Federico Becca, Saverio Moroni, and Stefano Baroni. Reptation quantum Monte Carlo algorithm for lattice Hamiltonians with a directed-update scheme. Physical Review E, 82 (4): 046710, 2010. 10.1103/PhysRevE.82.046710.
https://doi.org/10.1103/PhysRevE.82.046710

[12] Matthias Troyer and Uwe-Jens Wiese. Computational Complexity and Fundamental Limitations to Fermionic Quantum Monte Carlo Simulations. Physical Review Letters, 94 (17): 170201, 2005. 10.1103/PhysRevLett.94.170201.
https://doi.org/10.1103/PhysRevLett.94.170201

[13] Patrik Henelius and Anders W. Sandvik. Sign problem in Monte Carlo simulations of frustrated quantum spin systems. Physical Review B, 62 (2): 1102–1113, 2000. 10.1103/PhysRevB.62.1102.
https://doi.org/10.1103/PhysRevB.62.1102

[14] Philipp Hauke, Tommaso Roscilde, Valentin Murg, J. Ignacio Cirac, and Roman Schmied. Modified spin-wave theory with ordering vector optimization: Frustrated bosons on the spatially anisotropic triangular lattice. New Journal of Physics, 12 (5): 053036, 2010. 10.1088/1367-2630/12/5/053036.
https://doi.org/10.1088/1367-2630/12/5/053036

[15] Gaopei Pan and Zi Yang Meng. The sign problem in quantum Monte Carlo simulations. In Tapash Chakraborty, editor, Encyclopedia of Condensed Matter Physics (Second Edition), pages 879–893. Academic Press, Oxford, 2024. 10.1016/B978-0-323-90800-9.00095-0.
https://doi.org/10.1016/B978-0-323-90800-9.00095-0

[16] M. Ö. Oktel, M. Niţă, and B. Tanatar. Mean-field theory for Bose-Hubbard model under a magnetic field. Physical Review B, 75 (4): 045133, 2007. 10.1103/PhysRevB.75.045133.
https://doi.org/10.1103/PhysRevB.75.045133

[17] G. Möller and N. R. Cooper. Condensed ground states of frustrated Bose-Hubbard models. Physical Review A, 82 (6): 063625, 2010. 10.1103/PhysRevA.82.063625.
https://doi.org/10.1103/PhysRevA.82.063625

[18] Sebastian D. Huber and Netanel H. Lindner. Topological transitions for lattice bosons in a magnetic field. Proceedings of the National Academy of Sciences, 108 (50): 19925–19930, 2011. 10.1073/pnas.1110813108.
https://doi.org/10.1073/pnas.1110813108

[19] Arya Dhar, Maheswar Maji, Tapan Mishra, R. V. Pai, Subroto Mukerjee, and Arun Paramekanti. Bose-Hubbard model in a strong effective magnetic field: Emergence of a chiral Mott insulator ground state. Physical Review A, 85 (4): 041602, 2012. 10.1103/PhysRevA.85.041602.
https://doi.org/10.1103/PhysRevA.85.041602

[20] Arya Dhar, Tapan Mishra, Maheswar Maji, R. V. Pai, Subroto Mukerjee, and Arun Paramekanti. Chiral Mott insulator with staggered loop currents in the fully frustrated Bose-Hubbard model. Physical Review B, 87 (17): 174501, 2013. 10.1103/PhysRevB.87.174501.
https://doi.org/10.1103/PhysRevB.87.174501

[21] Akiyuki Tokuno and Antoine Georges. Ground states of a Bose–Hubbard ladder in an artificial magnetic field: Field-theoretical approach. New Journal of Physics, 16 (7): 073005, 2014. 10.1088/1367-2630/16/7/073005.
https://doi.org/10.1088/1367-2630/16/7/073005

[22] F. Kolley, M. Piraud, I. P. McCulloch, U. Schollwöck, and F. Heidrich-Meisner. Strongly interacting bosons on a three-leg ladder in the presence of a homogeneous flux. New Journal of Physics, 17 (9): 092001, 2015. 10.1088/1367-2630/17/9/092001.
https://doi.org/10.1088/1367-2630/17/9/092001

[23] Yin-Chen He, Subhro Bhattacharjee, R. Moessner, and Frank Pollmann. Bosonic Integer Quantum Hall Effect in an Interacting Lattice Model. Physical Review Letters, 115 (11): 116803, 2015. 10.1103/PhysRevLett.115.116803.
https://doi.org/10.1103/PhysRevLett.115.116803

[24] Christian Romen and Andreas M. Läuchli. Chiral Mott insulators in frustrated Bose-Hubbard models on ladders and two-dimensional lattices: A combined perturbative and density matrix renormalization group study. Physical Review B, 98 (5): 054519, 2018. 10.1103/PhysRevB.98.054519.
https://doi.org/10.1103/PhysRevB.98.054519

[25] Tian-Sheng Zeng, D. N. Sheng, and W. Zhu. Continuous phase transition between bosonic integer quantum Hall liquid and a trivial insulator: Evidence for deconfined quantum criticality. Physical Review B, 101 (3): 035138, 2020. 10.1103/PhysRevB.101.035138.
https://doi.org/10.1103/PhysRevB.101.035138

[26] Ya-Fei Song and Shi-Jie Yang. Quantum phases for bosons in a magnetic lattice with a harmonic trap. New Journal of Physics, 22 (7): 073001, 2020. 10.1088/1367-2630/ab95e0.
https://doi.org/10.1088/1367-2630/ab95e0

[27] Maximilian Buser, Claudius Hubig, Ulrich Schollwöck, Leticia Tarruell, and Fabian Heidrich-Meisner. Interacting bosonic flux ladders with a synthetic dimension: Ground-state phases and quantum quench dynamics. Physical Review A, 102 (5): 053314, 2020. 10.1103/PhysRevA.102.053314.
https://doi.org/10.1103/PhysRevA.102.053314

[28] Catalin-Mihai Halati and Thierry Giamarchi. Bose-Hubbard triangular ladder in an artificial gauge field. Physical Review Research, 5 (1): 013126, 2023. 10.1103/PhysRevResearch.5.013126.
https://doi.org/10.1103/PhysRevResearch.5.013126

[29] D. Jaksch and P. Zoller. Creation of effective magnetic fields in optical lattices: The Hofstadter butterfly for cold neutral atoms. New Journal of Physics, 5 (1): 56, 2003. 10.1088/1367-2630/5/1/356.
https://doi.org/10.1088/1367-2630/5/1/356

[30] Fabrice Gerbier and Jean Dalibard. Gauge fields for ultracold atoms in optical superlattices. New Journal of Physics, 12 (3): 033007, 2010. 10.1088/1367-2630/12/3/033007.
https://doi.org/10.1088/1367-2630/12/3/033007

[31] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch. Realization of the Hofstadter Hamiltonian with Ultracold Atoms in Optical Lattices. Physical Review Letters, 111 (18): 185301, 2013. 10.1103/PhysRevLett.111.185301.
https://doi.org/10.1103/PhysRevLett.111.185301

[32] Hirokazu Miyake, Georgios A. Siviloglou, Colin J. Kennedy, William Cody Burton, and Wolfgang Ketterle. Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices. Physical Review Letters, 111 (18): 185302, 2013. 10.1103/PhysRevLett.111.185302.
https://doi.org/10.1103/PhysRevLett.111.185302

[33] J. Struck, M. Weinberg, C. Ölschläger, P. Windpassinger, J. Simonet, K. Sengstock, R. Höppner, P. Hauke, A. Eckardt, M. Lewenstein, and L. Mathey. Engineering Ising-XY spin-models in a triangular lattice using tunable artificial gauge fields. Nature Physics, 9 (11): 738–743, 2013. 10.1038/nphys2750.
https://doi.org/10.1038/nphys2750

[34] Marcos Atala, Monika Aidelsburger, Michael Lohse, Julio T. Barreiro, Belén Paredes, and Immanuel Bloch. Observation of chiral currents with ultracold atoms in bosonic ladders. Nature Physics, 10 (8): 588–593, 2014. 10.1038/nphys2998.
https://doi.org/10.1038/nphys2998

[35] N. Goldman, G. Juzeliūnas, P. Öhberg, and I. B. Spielman. Light-induced gauge fields for ultracold atoms. Reports on Progress in Physics, 77 (12): 126401, 2014. 10.1088/0034-4885/77/12/126401.
https://doi.org/10.1088/0034-4885/77/12/126401

[36] A. Sterdyniak, Nigel R. Cooper, and N. Regnault. Bosonic Integer Quantum Hall Effect in Optical Flux Lattices. Physical Review Letters, 115 (11): 116802, 2015. 10.1103/PhysRevLett.115.116802.
https://doi.org/10.1103/PhysRevLett.115.116802

[37] Luca Barbiero, Josep Cabedo, Maciej Lewenstein, Leticia Tarruell, and Alessio Celi. Frustrated magnets without geometrical frustration in bosonic flux ladders. Physical Review Research, 5 (4): L042008, 2023. 10.1103/PhysRevResearch.5.L042008.
https://doi.org/10.1103/PhysRevResearch.5.L042008

[38] Manuela Capello, Federico Becca, Michele Fabrizio, Sandro Sorella, and Erio Tosatti. Variational Description of Mott Insulators. Physical Review Letters, 94 (2): 026406, 2005. 10.1103/PhysRevLett.94.026406.
https://doi.org/10.1103/PhysRevLett.94.026406

[39] Manuela Capello, Federico Becca, Michele Fabrizio, and Sandro Sorella. Superfluid to Mott-Insulator Transition in Bose-Hubbard Models. Physical Review Letters, 99 (5): 056402, 2007. 10.1103/PhysRevLett.99.056402.
https://doi.org/10.1103/PhysRevLett.99.056402

[40] Manuela Capello, Federico Becca, Michele Fabrizio, and Sandro Sorella. Mott transition in bosonic systems: Insights from the variational approach. Physical Review B, 77 (14): 144517, 2008. 10.1103/PhysRevB.77.144517.
https://doi.org/10.1103/PhysRevB.77.144517

[41] Vincenzo Alba, Masudul Haque, and Andreas M. Läuchli. Entanglement Spectrum of the Two-Dimensional Bose-Hubbard Model. Physical Review Letters, 110 (26): 260403, 2013. 10.1103/PhysRevLett.110.260403.
https://doi.org/10.1103/PhysRevLett.110.260403

[42] Marek M. Rams, Piotr Czarnik, and Lukasz Cincio. Precise Extrapolation of the Correlation Function Asymptotics in Uniform Tensor Network States with Application to the Bose-Hubbard and XXZ Models. Physical Review X, 8 (4): 041033, 2018. 10.1103/PhysRevX.8.041033.
https://doi.org/10.1103/PhysRevX.8.041033

[43] Erik Lennart Weerda and Matteo Rizzi. Fractional quantum Hall states with variational projected entangled-pair states: A study of the bosonic Harper-Hofstadter model. Physical Review B, 109 (24): L241117, 2024. 10.1103/PhysRevB.109.L241117.
https://doi.org/10.1103/PhysRevB.109.L241117

[44] J. Eisert, M. Cramer, and M. B. Plenio. Colloquium: Area laws for the entanglement entropy. Reviews of Modern Physics, 82 (1): 277–306, 2010. 10.1103/RevModPhys.82.277.
https://doi.org/10.1103/RevModPhys.82.277

[45] Ramesh V. Pai, Rahul Pandit, H. R. Krishnamurthy, and S. Ramasesha. One-Dimensional Disordered Bosonic Hubbard Model: A Density-Matrix Renormalization Group Study. Physical Review Letters, 76 (16): 2937–2940, 1996. 10.1103/PhysRevLett.76.2937.
https://doi.org/10.1103/PhysRevLett.76.2937

[46] T. D. Kühner and H. Monien. Phases of the one-dimensional Bose-Hubbard model. Physical Review B, 58 (22): R14741–R14744, 1998. 10.1103/PhysRevB.58.R14741.
https://doi.org/10.1103/PhysRevB.58.R14741

[47] S. Rapsch, U. Schollwöck, and W. Zwerger. Density matrix renormalization group for disordered bosons in one dimension. Europhysics Letters, 46 (5): 559, 1999. 10.1209/epl/i1999-00302-7.
https://doi.org/10.1209/epl/i1999-00302-7

[48] Corinna Kollath, Andreas M. Läuchli, and Ehud Altman. Quench Dynamics and Nonequilibrium Phase Diagram of the Bose-Hubbard Model. Physical Review Letters, 98 (18): 180601, 2007. 10.1103/PhysRevLett.98.180601.
https://doi.org/10.1103/PhysRevLett.98.180601

[49] Andreas M. Läuchli and Corinna Kollath. Spreading of correlations and entanglement after a quench in the one-dimensional Bose–Hubbard model. Journal of Statistical Mechanics: Theory and Experiment, 2008 (05): P05018, 2008. 10.1088/1742-5468/2008/05/P05018.
https://doi.org/10.1088/1742-5468/2008/05/P05018

[50] Giuseppe Carleo and Matthias Troyer. Solving the quantum many-body problem with artificial neural networks. Science, 355 (6325): 602–606, 2017. 10.1126/science.aag2302.
https://doi.org/10.1126/science.aag2302

[51] Yusuke Nomura and Masatoshi Imada. Dirac-Type Nodal Spin Liquid Revealed by Refined Quantum Many-Body Solver Using Neural-Network Wave Function, Correlation Ratio, and Level Spectroscopy. Physical Review X, 11 (3): 031034, 2021. 10.1103/PhysRevX.11.031034.
https://doi.org/10.1103/PhysRevX.11.031034

[52] Ao Chen and Markus Heyl. Empowering deep neural quantum states through efficient optimization. Nature Physics, 20 (9): 1476–1481, 2024. 10.1038/s41567-024-02566-1.
https://doi.org/10.1038/s41567-024-02566-1

[53] Riccardo Rende, Luciano Loris Viteritti, Lorenzo Bardone, Federico Becca, and Sebastian Goldt. A simple linear algebra identity to optimize large-scale neural network quantum states. Communications Physics, 7 (1): 260, 2024. 10.1038/s42005-024-01732-4.
https://doi.org/10.1038/s42005-024-01732-4

[54] Gabriel Pescia, Jiequn Han, Alessandro Lovato, Jianfeng Lu, and Giuseppe Carleo. Neural-network quantum states for periodic systems in continuous space. Physical Review Research, 4 (2): 023138, 2022. 10.1103/PhysRevResearch.4.023138.
https://doi.org/10.1103/PhysRevResearch.4.023138

[55] Jan Hermann, Zeno Schätzle, and Frank Noé. Deep-neural-network solution of the electronic Schrödinger equation. Nature Chemistry, 12 (10): 891–897, 2020. 10.1038/s41557-020-0544-y.
https://doi.org/10.1038/s41557-020-0544-y

[56] David Pfau, James S. Spencer, Alexander G. D. G. Matthews, and W. M. C. Foulkes. Ab initio solution of the many-electron Schrödinger equation with deep neural networks. Physical Review Research, 2 (3): 033429, 2020. 10.1103/PhysRevResearch.2.033429.
https://doi.org/10.1103/PhysRevResearch.2.033429

[57] Gabriel Pescia, Jannes Nys, Jane Kim, Alessandro Lovato, and Giuseppe Carleo. Message-passing neural quantum states for the homogeneous electron gas. Phys. Rev. B, 110: 035108, 2024. 10.1103/PhysRevB.110.035108.
https://doi.org/10.1103/PhysRevB.110.035108

[58] Di Luo, Giuseppe Carleo, Bryan K. Clark, and James Stokes. Gauge Equivariant Neural Networks for Quantum Lattice Gauge Theories. Physical Review Letters, 127 (27): 276402, 2021. 10.1103/PhysRevLett.127.276402.
https://doi.org/10.1103/PhysRevLett.127.276402

[59] Matija Medvidović and Dries Sels. Variational Quantum Dynamics of Two-Dimensional Rotor Models. PRX Quantum, 4 (4): 040302, 2023. 10.1103/PRXQuantum.4.040302.
https://doi.org/10.1103/PRXQuantum.4.040302

[60] Hiroki Saito. Solving the Bose–Hubbard Model with Machine Learning. Journal of the Physical Society of Japan, 86 (9): 093001, 2017. 10.7566/JPSJ.86.093001.
https://doi.org/10.7566/JPSJ.86.093001

[61] Kristopher McBrian, Giuseppe Carleo, and Ehsan Khatami. Ground state phase diagram of the one-dimensional Bose-Hubbard model from restricted Boltzmann machines. Journal of Physics: Conference Series, 1290 (1): 012005, 2019. 10.1088/1742-6596/1290/1/012005.
https://doi.org/10.1088/1742-6596/1290/1/012005

[62] Vladimir Vargas-Calderón, Herbert Vinck-Posada, and Fabio A. González. Phase Diagram Reconstruction of the Bose–Hubbard Model with a Restricted Boltzmann Machine Wavefunction. Journal of the Physical Society of Japan, 89 (9): 094002, 2020. 10.7566/JPSJ.89.094002.
https://doi.org/10.7566/JPSJ.89.094002

[63] Ziyan Zhu, Marios Mattheakis, Weiwei Pan, and Efthimios Kaxiras. HubbardNet: Efficient predictions of the Bose-Hubbard model spectrum with deep neural networks. Physical Review Research, 5 (4): 043084, 2023. 10.1103/PhysRevResearch.5.043084.
https://doi.org/10.1103/PhysRevResearch.5.043084

[64] Kenny Choo, Giuseppe Carleo, Nicolas Regnault, and Titus Neupert. Symmetries and Many-Body Excitations with Neural-Network Quantum States. Physical Review Letters, 121 (16): 167204, 2018. 10.1103/PhysRevLett.121.167204.
https://doi.org/10.1103/PhysRevLett.121.167204

[65] Michael Y Pei and Stephen R Clark. Specialising neural-network quantum states for the bose hubbard model. Journal of Physics B: Atomic, Molecular and Optical Physics, 57 (21): 215301, 2024. 10.1088/1361-6455/ad7e87.
https://doi.org/10.1088/1361-6455/ad7e87

[66] R. P. Feynman and Michael Cohen. Energy Spectrum of the Excitations in Liquid Helium. Physical Review, 102 (5): 1189–1204, 1956. 10.1103/PhysRev.102.1189.
https://doi.org/10.1103/PhysRev.102.1189

[67] Michael A. Lee, K. E. Schmidt, M. H. Kalos, and G. V. Chester. Green's Function Monte Carlo Method for Liquid ${}^{3}$He. Physical Review Letters, 46 (11): 728–731, 1981. 10.1103/PhysRevLett.46.728.
https://doi.org/10.1103/PhysRevLett.46.728

[68] Yongkyung Kwon, D. M. Ceperley, and Richard M. Martin. Effects of three-body and backflow correlations in the two-dimensional electron gas. Physical Review B, 48 (16): 12037–12046, 1993. 10.1103/PhysRevB.48.12037.
https://doi.org/10.1103/PhysRevB.48.12037

[69] Yongkyung Kwon, D. M. Ceperley, and Richard M. Martin. Effects of backflow correlation in the three-dimensional electron gas: Quantum Monte Carlo study. Physical Review B, 58 (11): 6800–6806, 1998. 10.1103/PhysRevB.58.6800.
https://doi.org/10.1103/PhysRevB.58.6800

[70] M. Holzmann, D. M. Ceperley, C. Pierleoni, and K. Esler. Backflow correlations for the electron gas and metallic hydrogen. Physical Review E, 68 (4): 046707, 2003. 10.1103/PhysRevE.68.046707.
https://doi.org/10.1103/PhysRevE.68.046707

[71] Luca F. Tocchio, Federico Becca, Alberto Parola, and Sandro Sorella. Role of backflow correlations for the nonmagnetic phase of the $t$–$t'$ Hubbard model. Physical Review B, 78 (4): 041101, 2008. 10.1103/PhysRevB.78.041101.
https://doi.org/10.1103/PhysRevB.78.041101

[72] Luca F. Tocchio, Federico Becca, and Claudius Gros. Backflow correlations in the Hubbard model: An efficient tool for the study of the metal-insulator transition and the large-$U$ limit. Physical Review B, 83 (19): 195138, 2011. 10.1103/PhysRevB.83.195138.
https://doi.org/10.1103/PhysRevB.83.195138

[73] Michele Ruggeri, Saverio Moroni, and Markus Holzmann. Nonlinear Network Description for Many-Body Quantum Systems in Continuous Space. Physical Review Letters, 120 (20): 205302, 2018. 10.1103/PhysRevLett.120.205302.
https://doi.org/10.1103/PhysRevLett.120.205302

[74] Di Luo and Bryan K. Clark. Backflow Transformations via Neural Networks for Quantum Many-Body Wave Functions. Physical Review Letters, 122 (22): 226401, 2019. 10.1103/PhysRevLett.122.226401.
https://doi.org/10.1103/PhysRevLett.122.226401

[75] Max A. Metlitski and Tarun Grover. Entanglement Entropy of Systems with Spontaneously Broken Continuous Symmetry, 2015.

[76] Irénée Frérot and Tommaso Roscilde. Entanglement Entropy across the Superfluid-Insulator Transition: A Signature of Bosonic Criticality. Physical Review Letters, 116 (19): 190401, 2016. 10.1103/PhysRevLett.116.190401.
https://doi.org/10.1103/PhysRevLett.116.190401

[77] Rajibul Islam, Ruichao Ma, Philipp M. Preiss, M. Eric Tai, Alexander Lukin, Matthew Rispoli, and Markus Greiner. Measuring entanglement entropy in a quantum many-body system. Nature, 528 (7580): 77–83, 2015. 10.1038/nature15750.
https://doi.org/10.1038/nature15750

[78] H. A. Gersch and G. C. Knollman. Quantum Cell Model for Bosons. Physical Review, 129 (2): 959–967, 1963. 10.1103/PhysRev.129.959.
https://doi.org/10.1103/PhysRev.129.959

[79] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller. Cold Bosonic Atoms in Optical Lattices. Physical Review Letters, 81 (15): 3108–3111, 1998. 10.1103/PhysRevLett.81.3108.
https://doi.org/10.1103/PhysRevLett.81.3108

[80] Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W. Hänsch, and Immanuel Bloch. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature, 415 (6867): 39–44, 2002. 10.1038/415039a.
https://doi.org/10.1038/415039a

[81] I. B. Spielman, W. D. Phillips, and J. V. Porto. Mott-Insulator Transition in a Two-Dimensional Atomic Bose Gas. Physical Review Letters, 98 (8): 080404, 2007. 10.1103/PhysRevLett.98.080404.
https://doi.org/10.1103/PhysRevLett.98.080404

[82] I. B. Spielman, W. D. Phillips, and J. V. Porto. Condensate Fraction in a 2D Bose Gas Measured across the Mott-Insulator Transition. Physical Review Letters, 100 (12): 120402, 2008. 10.1103/PhysRevLett.100.120402.
https://doi.org/10.1103/PhysRevLett.100.120402

[83] Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger. Many-body physics with ultracold gases. Reviews of Modern Physics, 80 (3): 885–964, 2008. 10.1103/RevModPhys.80.885.
https://doi.org/10.1103/RevModPhys.80.885

[84] Manuel Endres, Takeshi Fukuhara, David Pekker, Marc Cheneau, Peter Schauß, Christian Gross, Eugene Demler, Stefan Kuhr, and Immanuel Bloch. The `Higgs' amplitude mode at the two-dimensional superfluid/Mott insulator transition. Nature, 487 (7408): 454–458, 2012. 10.1038/nature11255.
https://doi.org/10.1038/nature11255

[85] A. van Otterlo, K.-H. Wagenblast, Rosario Fazio, and Gerd Schön. Response of Josephson-junction arrays near the quantum phase transition. Physical Review B, 48 (5): 3316–3326, 1993. 10.1103/PhysRevB.48.3316.
https://doi.org/10.1103/PhysRevB.48.3316

[86] C. Bruder, R. Fazio, and G. Schön. The Bose-Hubbard model: from Josephson junction arrays to optical lattices. Annalen der Physik, 517 (9-10): 566–577, 2005. 10.1002/andp.200551709-1005.
https://doi.org/10.1002/andp.200551709-1005

[87] C. N. Yang. Concept of Off-Diagonal Long-Range Order and the Quantum Phases of Liquid He and of Superconductors. Reviews of Modern Physics, 34 (4): 694–704, 1962. 10.1103/RevModPhys.34.694.
https://doi.org/10.1103/RevModPhys.34.694

[88] Matthew P. A. Fisher, Peter B. Weichman, G. Grinstein, and Daniel S. Fisher. Boson localization and the superfluid-insulator transition. Physical Review B, 40 (1): 546–570, 1989. 10.1103/PhysRevB.40.546.
https://doi.org/10.1103/PhysRevB.40.546

[89] W. Krauth and N. Trivedi. Mott and Superfluid Transitions in a Strongly Interacting Lattice Boson System. Europhysics Letters, 14 (7): 627, 1991. 10.1209/0295-5075/14/7/003.
https://doi.org/10.1209/0295-5075/14/7/003

[90] J. K. Freericks and H. Monien. Strong-coupling expansions for the pure and disordered Bose-Hubbard model. Physical Review B, 53 (5): 2691–2700, 1996. 10.1103/PhysRevB.53.2691.
https://doi.org/10.1103/PhysRevB.53.2691

[91] N. Elstner and H. Monien. Dynamics and thermodynamics of the Bose-Hubbard model. Physical Review B, 59 (19): 12184–12187, 1999. 10.1103/PhysRevB.59.12184.
https://doi.org/10.1103/PhysRevB.59.12184

[92] A. Bijl. The lowest wave function of the symmetrical many particles system. Physica, 7 (9): 869–886, 1940. 10.1016/0031-8914(40)90166-5.
https://doi.org/10.1016/0031-8914(40)90166-5

[93] Robert Jastrow. Many-Body Problem with Strong Forces. Physical Review, 98 (5): 1479–1484, 1955. 10.1103/PhysRev.98.1479.
https://doi.org/10.1103/PhysRev.98.1479

[94] Luciano Loris Viteritti, Riccardo Rende, and Federico Becca. Transformer Variational Wave Functions for Frustrated Quantum Spin Systems. Physical Review Letters, 130 (23): 236401, 2023. 10.1103/PhysRevLett.130.236401.
https://doi.org/10.1103/PhysRevLett.130.236401

[95] Luciano Loris Viteritti, Riccardo Rende, Alberto Parola, Sebastian Goldt, and Federico 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. 10.1103/PhysRevB.111.134411.
https://doi.org/10.1103/PhysRevB.111.134411

[96] Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation Learning: A Review and New Perspectives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (8): 1798–1828, 2013. 10.1109/TPAMI.2013.50.
https://doi.org/10.1109/TPAMI.2013.50

[97] Sandro Sorella. Wave function optimization in the variational Monte Carlo method. Physical Review B, 71 (24): 241103, 2005. 10.1103/PhysRevB.71.241103.
https://doi.org/10.1103/PhysRevB.71.241103

[98] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo. Quantum Natural Gradient. Quantum, 4: 269, 2020. 10.22331/q-2020-05-25-269.
https://doi.org/10.22331/q-2020-05-25-269

[99] Filippo Vicentini, Damian Hofmann, Attila Szabó, Dian Wu, Christopher Roth, Clemens Giuliani, Gabriel Pescia, Jannes Nys, Vladimir Vargas-Calderón, Nikita Astrakhantsev, and Giuseppe Carleo. NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems. SciPost Physics Codebases, page 007, 2022. 10.21468/SciPostPhysCodeb.7.
https://doi.org/10.21468/SciPostPhysCodeb.7

[100] Federico Becca. Private communication, 2023.

[101] Dian Wu, Riccardo Rossi, Filippo Vicentini, Nikita Astrakhantsev, Federico Becca, Xiaodong Cao, Juan Carrasquilla, Francesco Ferrari, Antoine Georges, Mohamed Hibat-Allah, Masatoshi Imada, Andreas M. Läuchli, Guglielmo Mazzola, Antonio Mezzacapo, Andrew Millis, Javier Robledo Moreno, Titus Neupert, Yusuke Nomura, Jannes Nys, Olivier Parcollet, Rico Pohle, Imelda Romero, Michael Schmid, J. Maxwell Silvester, Sandro Sorella, Luca F. Tocchio, Lei Wang, Steven R. White, Alexander Wietek, Qi Yang, Yiqi Yang, Shiwei Zhang, and Giuseppe Carleo. Variational benchmarks for quantum many-body problems. Science, 386 (6719): 296–301, 2024. 10.1126/science.adg9774.
https://doi.org/10.1126/science.adg9774

[102] Massimo Campostrini, Martin Hasenbusch, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari. Critical behavior of the three-dimensional XY universality class. Physical Review B, 63 (21): 214503, 2001. 10.1103/PhysRevB.63.214503.
https://doi.org/10.1103/PhysRevB.63.214503

[103] Matthew B. Hastings, Iván González, Ann B. Kallin, and Roger G. Melko. Measuring Renyi Entanglement Entropy in Quantum Monte Carlo Simulations. Physical Review Letters, 104 (15): 157201, 2010. 10.1103/PhysRevLett.104.157201.
https://doi.org/10.1103/PhysRevLett.104.157201

[104] Giacomo Torlai, Guglielmo Mazzola, Juan Carrasquilla, Matthias Troyer, Roger Melko, and Giuseppe Carleo. Neural-network quantum state tomography. Nature Physics, 14 (5): 447–450, 2018. 10.1038/s41567-018-0048-5.
https://doi.org/10.1038/s41567-018-0048-5

[105] Jiarui Zhao, Bin-Bin Chen, Yan-Cheng Wang, Zheng Yan, Meng Cheng, and Zi Yang Meng. Measuring Rényi entanglement entropy with high efficiency and precision in quantum Monte Carlo simulations. npj Quantum Materials, 7 (1): 1–9, 2022. 10.1038/s41535-022-00476-0.
https://doi.org/10.1038/s41535-022-00476-0

[106] Zakari Denis. cqsl/lattice-bosons: v1.0.0, June 2025. URL https://doi.org/10.5281/zenodo.15632176.
https://doi.org/10.5281/zenodo.15632176

Cited by

[1] Zihao Qi and Christopher Earls, "Attention in Krylov space: Transformer-based extrapolation of Lanczos coefficients", Physical Review B 114 1, 014319 (2026).

[2] Waleed Sherif, "Simultaneous approximation of multiple degenerate states using a single neural network quantum state", Machine Learning: Science and Technology 7 1, 015029 (2026).

[3] Rajah P. Nutakki, Ahmedeo Shokry, and Filippo Vicentini, "Design principles of deep translationally symmetric neural quantum states for frustrated magnets", Physical Review Research 7 4, 043099 (2025).

[4] Alessandro Sinibaldi, Antonio Francesco Mello, Mario Collura, and Giuseppe Carleo, "Nonstabilizerness of neural quantum states", Physical Review Research 7 4, 043289 (2025).

[5] Marco Julian Solanki, Lexin Ding, and Markus Reiher, "Neural Quantum States Based on Selected Configurations", The Journal of Physical Chemistry Letters 17 18, 5180 (2026).

[6] Rakesh Sengupta, "Mapping additive recurrent neural networks to quantum neural networks", Quantum Information Processing 24 6, 185 (2025).

[7] Adrien Kahn, Luca Gravina, and Filippo Vicentini, "Variational subspace methods and application to improving variational Monte Carlo dynamics", Quantum 10, 2082 (2026).

[8] Ronald Santiago Cortes Santamaria, Aravindh S. Shankar, Marcello Dalmonte, Roberto Verdel, and Nils Niggemann, "Basis dependence of neural quantum states for the transverse field Ising model", SciPost Physics Core 9 2, 027 (2026).

[9] Riccardo Rende and Luciano Loris Viteritti, "Are queries and keys always relevant? A case study on transformer wave functions", Machine Learning: Science and Technology 6 1, 010501 (2025).

[10] Fabian Döschl, Felix A. Palm, Hannah Lange, Fabian Grusdt, and Annabelle Bohrdt, "Neural network quantum states for the interacting Hofstadter model with higher local occupations and long-range interactions", Physical Review B 111 4, 045408 (2025).

[11] Björn J. Wurst, Dante M. Kennes, and Jonas B. Profe, "Efficiency of the hidden fermion determinant states Ansatz in the light of different complexity measures", Physical Review Research 7 2, 023316 (2025).

[12] Douglas Hendry, Alessandro Sinibaldi, and Giuseppe Carleo, "Grassmann Variational Monte Carlo with neural wave functions", arXiv:2507.10287, (2025).

[13] Ronald Santiago Cortes, Aravindh S. Shankar, Marcello Dalmonte, Roberto Verdel, and Nils Niggemann, "Basis dependence of Neural Quantum States for the Transverse Field Ising Model", arXiv:2512.11632, (2025).

[14] Marc Machaczek, Lode Pollet, and Ke Liu, "Neural quantum state study of fracton models", SciPost Physics 18 3, 112 (2025).

[15] Marc Machaczek, Lode Pollet, and Ke Liu, "Neural Quantum State Study of Fracton Models", arXiv:2406.11677, (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-07 18:54:47) and SAO/NASA ADS (last updated successfully 2026-08-07 18:54:50). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.