pith. sign in

arxiv: 2606.02794 · v1 · pith:2LPWHW7Onew · submitted 2026-06-01 · ❄️ cond-mat.dis-nn · cond-mat.str-el· cs.CC· quant-ph

Scaling Laws for Neural-Network Quantum States

Riccardo Rende , Alessandro Sinibaldi , Luciano Loris Viteritti , Roeland Wiersema , Antoine Georges , Giuseppe Carleo This is my paper

Pith reviewed 2026-06-28 11:28 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.str-elcs.CCquant-ph
keywords scaling lawsneural-network quantum statesvariational methodsJ1-J2 Heisenberg modeltransformer wave functionsV-scorefrustration
0
0 comments X

The pith

Transformer wave functions for quantum spin models improve in accuracy as a power law with training compute.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether scaling laws familiar from machine learning can describe the difficulty of approximating quantum ground states. For the J1-J2 Heisenberg model on square and triangular lattices, a transformer-based variational state shows that its V-score, a measure of accuracy, falls as a power law when more compute is spent on training. Results for lattices of different sizes collapse onto the same curve after rescaling compute, and the power law itself stays roughly the same regardless of system size. The rate of improvement slows as frustration increases, turning the exponent into a direct indicator of how hard the ground state is to represent.

Core claim

Transformer wave functions approximate ground states of the J1-J2 Heisenberg model on lattices with up to 20 by 20 sites. The V-score decays as a power law in training compute, with results for different system sizes collapsing onto one curve after rescaling compute. The power law is approximately independent of the number of sites, and its exponent decreases with frustration, quantifying the representational difficulty of the ground state.

What carries the argument

The V-score, a measure of accuracy of a variational state, which decays as a power law in training compute with data collapse under rescaling.

If this is right

  • The transformer Ansatz is size-consistent for the systems considered.
  • The exponent of the power law serves as a quantitative measure of representational difficulty.
  • Scaling laws provide a framework for benchmarking different variational ansatze against one another.
  • Additional training compute improves accuracy at a rate set by the exponent, largely independent of lattice size.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same scaling analysis could be applied to other variational architectures to compare their efficiency on the same physical models.
  • The dependence of the exponent on frustration may connect to physical features such as entanglement structure or the sign problem.
  • Extending the study to three-dimensional lattices or to models with longer-range interactions would test whether the size-independence persists.

Load-bearing premise

The observed power-law scaling, data collapse, and frustration dependence are not artifacts of the particular transformer architecture, optimization schedule, or definition of the V-score.

What would settle it

A clear deviation from power-law decay or failure of data collapse when repeating the calculations with a different neural-network architecture or optimization method would falsify the generality of the scaling.

Figures

Figures reproduced from arXiv: 2606.02794 by Alessandro Sinibaldi, Antoine Georges, Giuseppe Carleo, Luciano Loris Viteritti, Riccardo Rende, Roeland Wiersema.

Figure 1
Figure 1. Figure 1: FIG. 1. Scaling collapse of the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Illustration of the scaling collapse procedure for the square [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Cost [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
read the original abstract

Scaling laws, the power-law relations between loss, architecture size, and compute observed in modern neural networks, offer a quantitative way to characterize the complexity of a learning problem, with the exponent governing the decay of the loss reflecting how rapidly additional resources translate into improved accuracy, and thus how hard the target is to learn. Whether an analogous framework can characterize the complexity of physical problems remains open. We address this question for Neural-Network Quantum States, a leading variational approach for strongly correlated quantum many-body systems. Using transformer wave functions to approximate ground states of the $J_1$-$J_2$ Heisenberg model on triangular and square lattices with up to $20\times 20$ sites, we find that the $V$-score, a measure of accuracy of a variational state, decays as a power law in training compute. Under an appropriate rescaling of compute, results for different system sizes collapse onto a single curve, analogous to scaling collapse in critical phenomena. The resulting power law is, to a good approximation, independent of the number of sites, showing that the transformer Ansatz is size-consistent for the systems considered. The exponent decreases systematically with frustration, identifying it as a quantitative measure of representational difficulty of the ground state and establishing scaling laws as a general framework for benchmarking variational ans\"{a}tze.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper reports empirical observations using transformer-based neural-network quantum states to approximate ground states of the J1-J2 Heisenberg model on square and triangular lattices (up to 20x20 sites). It finds that the V-score decays as a power law in training compute, that results for different system sizes collapse onto a single curve under appropriate compute rescaling, that the power law is approximately independent of system size, and that the fitted exponent decreases systematically with increasing frustration. These are presented as establishing scaling laws as a general framework for benchmarking variational ansätze and quantifying representational difficulty of ground states.

Significance. If the reported power-law scaling, size collapse, and frustration dependence prove robust, they would offer a quantitative, resource-based diagnostic for the difficulty of variational ground-state approximation in quantum many-body systems, analogous to scaling laws in machine learning. The size-consistency observation and the link between exponent and frustration could help compare ansätze and identify regimes where additional compute yields diminishing returns.

major comments (2)
  1. [Abstract] Abstract: The claim that the exponent 'identifies it as a quantitative measure of representational difficulty of the ground state' and that the results establish 'scaling laws as a general framework for benchmarking variational ansätze' rests on the assumption that the observed scaling and frustration dependence are properties of the learning problem rather than the specific transformer architecture; all reported results use only transformer wave functions, with no ablations or comparisons to other NNQS forms (e.g., RBM, CNN, or MLP) described, which is load-bearing for the generality interpretation.
  2. [Methods] Methods (implied by abstract description of fits and collapse): No information is provided on error bars for V-score values, number of independent runs per data point, criteria for excluding optimization trajectories, or robustness checks against hyperparameter variation and optimizer stochasticity; without these, the reliability of the claimed power-law exponents and data collapse cannot be assessed.
minor comments (1)
  1. [Abstract] Abstract: The range of system sizes entering the collapse analysis is not stated explicitly (only the maximum 20x20 is given), which would help readers evaluate the collapse claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for these constructive comments, which highlight important aspects of our claims and presentation. We address each point below and will make corresponding revisions to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the exponent 'identifies it as a quantitative measure of representational difficulty of the ground state' and that the results establish 'scaling laws as a general framework for benchmarking variational ansätze' rests on the assumption that the observed scaling and frustration dependence are properties of the learning problem rather than the specific transformer architecture; all reported results use only transformer wave functions, with no ablations or comparisons to other NNQS forms (e.g., RBM, CNN, or MLP) described, which is load-bearing for the generality interpretation.

    Authors: We agree that the reported results are obtained exclusively with transformer wave functions and that no direct comparisons to other NNQS architectures are provided. The abstract language framing the findings as establishing 'scaling laws as a general framework' therefore exceeds what the data strictly support. In the revised manuscript we will rephrase the abstract to state that the observed power-law scaling, size collapse, and frustration dependence are demonstrated for transformer-based NNQS on the J1-J2 model, and that these results suggest scaling laws may offer a useful benchmarking approach; we will explicitly note that extension to other ansätze remains an open question for future work. This revision removes the overclaim while preserving the core empirical observations. revision: yes

  2. Referee: [Methods] Methods (implied by abstract description of fits and collapse): No information is provided on error bars for V-score values, number of independent runs per data point, criteria for excluding optimization trajectories, or robustness checks against hyperparameter variation and optimizer stochasticity; without these, the reliability of the claimed power-law exponents and data collapse cannot be assessed.

    Authors: We accept that the current manuscript lacks these statistical details. In the revised version we will add a dedicated subsection in Methods reporting: (i) the number of independent training runs performed for each system size and frustration value (typically five or more), (ii) error bars on V-score values computed from the standard deviation across runs, (iii) the convergence criteria used to retain or discard trajectories (e.g., energy stabilization within a tolerance over a fixed number of steps), and (iv) a brief statement on robustness to moderate hyperparameter changes. These additions will allow readers to evaluate the reliability of the reported exponents and collapse. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical power-law fits to observed V-score data

full rationale

The paper reports numerical experiments training transformer wave functions on J1-J2 Heisenberg models and fitting observed V-score decay versus compute. No derivation chain, uniqueness theorem, or self-citation is invoked to obtain the power laws, data collapse, or frustration dependence; these are direct empirical measurements. The V-score itself is an independent accuracy metric, and the reported exponents are statistical fits to training curves rather than quantities defined in terms of those same fits. No step reduces by construction to its inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

The central claim rests on empirical fitting of power-law exponents to V-score versus compute data for a specific neural architecture and model; no additional free parameters, axioms, or invented entities are introduced beyond standard variational Monte Carlo assumptions.

free parameters (1)
  • scaling exponent alpha
    The power-law exponent governing V-score decay is extracted by fitting to training curves at different frustration values.

pith-pipeline@v0.9.1-grok · 5787 in / 1280 out tokens · 34288 ms · 2026-06-28T11:28:22.442383+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

86 extracted references · 5 canonical work pages

  1. [1]

    Kaplan, S

    J. Kaplan, S. McCandlish, T. Henighan, T. B. Brown, B. Chess, R. Child, S. Gray, A. Radford, J. Wu, and D. Amodei, Scaling laws for neural language models (2020), arXiv:2001.08361 [cs.LG]

    Pith/arXiv arXiv 2020

  2. [2]

    Hoffmann, S

    J. Hoffmann, S. Borgeaud, A. Mensch, E. Buchatskaya, T. Cai, E. Rutherford, D. de Las Casas, L. A. Hendricks, J. Welbl, A. Clark, T. Hennigan, E. Noland, K. Milli- can, G. van den Driessche, B. Damoc, A. Guy, S. Osin- dero, K. Simonyan, E. Elsen, J. W. Rae, O. Vinyals, and L.Sifre,Trainingcompute-optimallargelanguagemodels (2022), arXiv:2203.15556 [cs.CL]

    Pith/arXiv arXiv 2022

  3. [3]

    X. Zhai, A. Kolesnikov, N. Houlsby, and L. Beyer, Scaling vision transformers (2022), arXiv:2106.04560 [cs.CV]

    arXiv 2022

  4. [4]

    Dehghani, J

    M. Dehghani, J. Djolonga, B. Mustafa, P. Padlewski, J. Heek, J. Gilmer, A. Steiner, M. Caron, R. Geirhos, I. Alabdulmohsin, R. Jenatton, L. Beyer, M. Tschan- nen, A. Arnab, X. Wang, C. Riquelme, M. Minderer, J. Puigcerver, U. Evci, M. Kumar, S. van Steenkiste, G. F. Elsayed, A. Mahendran, F. Yu, A. Oliver, F. Huot, J. Bastings, M. P. Collier, A. Gritsenko...

    arXiv 2023

  5. [5]

    Alabdulmohsin, X

    I. Alabdulmohsin, X. Zhai, A. Kolesnikov, and L. Beyer, Getting vit in shape: Scaling laws for compute-optimal model design (2024), arXiv:2305.13035 [cs.CV]

    arXiv 2024

  6. [6]

    Henighan, J

    T. Henighan, J. Kaplan, M. Katz, M. Chen, C. Hesse, J. Jackson, H. Jun, T. B. Brown, P. Dhariwal, S. Gray, C. Hallacy, B. Mann, A. Radford, A. Ramesh, N. Ryder, D. M. Ziegler, J. Schulman, D. Amodei, and S. McCan- dlish, Scaling laws for autoregressive generative modeling (2020), arXiv:2010.14701 [cs.LG]

    Pith/arXiv arXiv 2020

  7. [7]

    Hilton, J

    J. Hilton, J. Tang, and J. Schulman, Scaling laws for single-agent reinforcement learning (2023), arXiv:2301.13442 [cs.LG]

    arXiv 2023

  8. [8]

    Neumann and C

    O. Neumann and C. Gros, Scaling laws for a multi-agent reinforcement learning model (2022)

    2022

  9. [9]

    Cagnetta, A

    F. Cagnetta, A. Raventós, S. Ganguli, and M. Wyart, Deriving neural scaling laws from the statistics of natural language (2026), arXiv:2602.07488 [cs.LG]

    arXiv 2026

  10. [10]

    Bahri, E

    Y. Bahri, E. Dyer, J. Kaplan, J. Lee, and U. Sharma, Ex- plaining neural scaling laws, Proceedings of the National Academy of Sciences121, e2311878121 (2024)

    2024

  11. [11]

    L. L. Viteritti, R. Rende, and F. Becca, Transformer vari- ational wave functions for frustrated quantum spin sys- tems, Phys. Rev. Lett.130, 236401 (2023)

    2023

  12. [12]

    J. A. Sobral, M. Perle, and M. S. Scheurer, Physics- informed transformers for electronic quantum states, Nature Communications16, 10.1038/s41467-025-66844-z (2025)

    work page doi:10.1038/s41467-025-66844-z 2025

  13. [13]

    R. P. Nutakki and F. Vicentini, Ground-state phases of s= 1/2heisenberg models on the body-centered cubic lattice (2026), arXiv:2602.10008 [cond-mat.str-el]

    arXiv 2026

  14. [14]

    Raikos, S

    A. Raikos, S. Capponi, and F. Alet, Variational study of the magnetization plateaus in the spin- 1/2 kagome heisenberg antiferromagnet: an approach from vision transformer neural quantum states (2026), arXiv:2602.12998 [cond-mat.str-el]

    arXiv 2026

  15. [15]

    Yamazaki, I

    K. Yamazaki, I. Sakata, T. Konishi, and Y. Kawa- hara, Physics-inspired transformer quantum states via la- tent imaginary-time evolution (2026), arXiv:2602.03031 [cond-mat.dis-nn]

    arXiv 2026

  16. [17]

    L. L. Viteritti, R. Rende, G. Bracci-Testasecca, J. Niedda, R. Moessner, G. Carleo, and A. Scardicchio, Quantum spin glass in the two-dimensional disordered heisenberg model via foundation neural-network quan- tum states (2026), arXiv:2507.05073 [cond-mat.dis-nn]

    arXiv 2026

  17. [18]

    von Glehn, J

    I. von Glehn, J. S. Spencer, and D. Pfau, A self- attention ansatz for ab-initio quantum chemistry (2023), arXiv:2211.13672 [physics.chem-ph]

    arXiv 2023

  18. [19]

    Shang, C

    H. Shang, C. Guo, Y. Wu, Z. Li, and J. Yang, Solving the many-electron Schrödinger equation with a transformer- based framework, Nature Communications16, 8464 (2025)

    2025

  19. [20]

    M.E.MswahiliandY.-S.Jeong,Transformer-basedmod- els for chemical smiles representation: A comprehensive literature review, Heliyon10, e39038 (2024)

    2024

  20. [21]

    E. O. Pyzer-Knapp, M. Manica, P. Staar, L. Morin, P. Ruch, T. Laino, J. R. Smith, and A. Curioni, Founda- tion models for materials discovery – current state and future directions, npj Computational Materials11, 61 (2025)

    2025

  21. [22]

    Jumper, R

    J. Jumper, R. Evans, A. Pritzel, T. Green, M. Fig- urnov, O. Ronneberger, K. Tunyasuvunakool, R. Bates, A. Žídek, A. Potapenko, A. Bridgland, C. Meyer, S. A. A. Kohl, A. J. Ballard, A. Cowie, B. Romera-Paredes, S. Nikolov, R. Jain, J. Adler, T. Back, S. Petersen, D. Reiman, E. Clancy, M. Zielinski, M. Steinegger, M. Pacholska, T. Berghammer, S. Bodenstein...

    2021

  22. [23]

    Moussad, R

    B. Moussad, R. Roche, and D. Bhattacharya, The trans- formative power of transformers in protein structure pre- diction, Proceedings of the National Academy of Sciences 120, e2303499120 (2023)

    2023

  23. [24]

    Carleo, M

    G. Carleo and M. Troyer, Solving the quan- tum many-body problem with artificial neu- ral networks, Science355, 602 (2017), https://www.science.org/doi/pdf/10.1126/science.aag2302

    work page doi:10.1126/science.aag2302 2017

  24. [25]

    Becca and S

    F. Becca and S. Sorella,Quantum Monte Carlo Ap- proaches for Correlated Systems(Cambridge University Press, 2017)

    2017

  25. [26]

    H.Lange, A.VandeWalle, A.Abedinnia,andA.Bohrdt, From architectures to applications: a review of neural quantum states, Quantum Science and Technology9, 040501 (2024)

    2024

  26. [27]

    K. Choo, T. Neupert, and G. Carleo, Two-dimensional frustratedJ 1−J2 model studied with neural network quantum states, Phys. Rev. B100, 125124 (2019)

    2019

  27. [28]

    Chen and M

    A. Chen and M. Heyl, Empowering deep neural quantum states through efficient optimization, Nature Physics20, 1476 (2024)

    2024

  28. [29]

    A. Chen, V. D. Naik, and M. Heyl, Convolutional trans- former wave functions (2025), arXiv:2503.10462 [cond- mat.dis-nn]

    arXiv 2025

  29. [30]

    Hendry, A

    D. Hendry, A. Sinibaldi, and G. Carleo, Grassmann vari- 9 ational monte carlo with neural wave functions (2025), arXiv:2507.10287 [quant-ph]

    arXiv 2025

  30. [31]

    Hibat-Allah, M

    M. Hibat-Allah, M. Ganahl, L. E. Hayward, R. G. Melko, and J. Carrasquilla, Recurrent neural network wave func- tions, Phys. Rev. Res.2, 023358 (2020)

    2020

  31. [32]

    M. S. Moss, R. Wiersema, M. Hibat-Allah, J. Car- rasquilla, and R. G. Melko, Leveraging recurrence in neural network wavefunctions for large-scale simulations of heisenberg antiferromagnets on the triangular lattice, Phys. Rev. B112, 134449 (2025)

    2025

  32. [33]

    O.Sharir, Y.Levine, N.Wies, G.Carleo,andA.Shashua, Deep autoregressive models for the efficient variational simulation of many-body quantum systems, Phys. Rev. Lett.124, 020503 (2020)

    2020

  33. [34]

    Sprague and S

    K. Sprague and S. Czischek, Variational monte carlo with large patched transformers, Communications Physics7, 90 (2024)

    2024

  34. [35]

    N.-C., Pei, T., Liu, Z., Wang, X., Ning, G., Chan, T

    R. Rende, L. L. Viteritti, F. Becca, A. Scardicchio, A. Laio, and G. Carleo, Foundation neural-networks quantum states as a unified ansatz for multiple hamilto- nians, Nature Communications16, 10.1038/s41467-025- 62098-x (2025)

    work page doi:10.1038/s41467-025- 2025

  35. [36]

    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. B111, 134411 (2025)

    2025

  36. [37]

    C. Fan, B. Zhan, Y. Gu, T. Liu, Y. Wu, M. Qin, D. Lv, and T. Xiang, Disentangling tensor network states with deep neural network (2026), arXiv:2603.14425 [cond- mat.str-el]

    arXiv 2026

  37. [38]

    L. L. Viteritti, R. Rende, S. Sachdev, and G. Carleo, Ap- proaching the thermodynamic limit with neural-network quantumstates,arXivpreprintarXiv:2602.02665 (2026)

    arXiv 2026

  38. [39]

    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 (2025), arXiv:2507.02644 [cond-mat.str-el]

    arXiv 2025

  39. [40]

    L. L. Viteritti, R. Rende, C. Roth, A. Sengupta, G. Car- leo, and A. Georges, Beyond variational bias: Resolv- ing intertwined orders in the hubbard model (2026), arXiv:2604.21978 [cond-mat.str-el]

    Pith/arXiv arXiv 2026

  40. [41]

    Rende, A

    R. Rende, A. Nikolaenko, L. L. Viteritti, S. Sachdev, and Y.-H. Zhang, Transformer neural-network quantum states for lattice models of spins and fermions: Applica- tion to the ancilla layer model (2026), arXiv:2603.02316 [cond-mat.str-el]

    arXiv 2026

  41. [42]

    X. Cao, Z. Zhong, and Y. Lu, Vision transformer neural quantum states for impurity models (2024), arXiv:2408.13050 [cond-mat.str-el]

    arXiv 2024

  42. [43]

    Shang, C

    H. Shang, C. Guo, Y. Wu, Z. Li, and J. Yang, Solving the many-electron schrödinger equation with a transformer- based framework, Nature Communications16, 8464 (2025)

    2025

  43. [44]

    T. Zhao, J. Stokes, and S. Veerapaneni, Scalable neu- ral quantum states architecture for quantum chemistry, Machine Learning: Science and Technology4, 025034 (2023)

    2023

  44. [45]

    Knitter, D

    O. Knitter, D. Zhao, S. Leichenauer, and S. Veerapaneni, Large language model scaling laws for neural quantum states in quantum chemistry (2025), arXiv:2509.12679 [cs.LG]

    arXiv 2025

  45. [46]

    Y. Lu, S. Bharadwaj, D. Rathore, and D. Luo, Information-theoretic scaling laws of neural quantum states (2026), arXiv:2603.23468 [quant-ph]

    arXiv 2026

  46. [47]

    D. Wu, R. Rossi, F. Vicentini, N. Astrakhantsev, F. Becca, X. Cao, J. Carrasquilla, F. Ferrari, A. Georges, M. Hibat-Allah,et al., Variational benchmarks for quan- tum many-body problems, Science386, 296 (2024)

    2024

  47. [48]

    Pearce and J

    T. Pearce and J. Song, Reconciling kaplan and chinchilla scaling laws (2024), arXiv:2406.12907 [cs.LG]

    arXiv 2024

  48. [49]

    H. Li, W. Zheng, Q. Wang, Z. Ding, H. Wang, Z. Wang, S. Xuyang, N. Ding, S. Zhou, X. Zhang, and D. Jiang, Predictablescale: Partii, farseer: Arefinedscalinglawin large language models (2025), arXiv:2506.10972 [cs.LG]

    arXiv 2025

  49. [50]

    A. W. Sandvik, Finite-size scaling of the ground-state pa- rameters of the two-dimensional heisenberg model, Phys. Rev. B56, 11678 (1997)

    1997

  50. [51]

    A. W. Sandvik, High-precision ground state parameters of the two-dimensional spin-1/2 heisenberg model on the square lattice, Journal of Statistical Mechanics: Theory and Experiment2026, 043101 (2026)

    2026

  51. [52]

    Calandra Buonaura and S

    M. Calandra Buonaura and S. Sorella, Numerical study of the two-dimensional heisenberg model using a green function monte carlo technique with a fixed number of walkers, Phys. Rev. B57, 11446 (1998)

    1998

  52. [53]

    W.-J. Hu, F. Becca, A. Parola, and S. Sorella, Direct evidence for a gaplessZ2 spin liquid by frustrating néel antiferromagnetism, Phys. Rev. B88, 060402(R) (2013)

    2013

  53. [54]

    Nomura and M

    Y. Nomura and M. Imada, Dirac-type nodal spin liq- uid revealed by refined quantum many-body solver using neural-network wave function, correlation ratio, and level spectroscopy, Phys. Rev. X11, 031034 (2021)

    2021

  54. [55]

    Capriotti, A

    L. Capriotti, A. E. Trumper, and S. Sorella, Long-range néel order in the triangular heisenberg model, Phys. Rev. Lett.82, 3899 (1999)

    1999

  55. [56]

    S. R. White and A. L. Chernyshev, Neél order in square and triangular lattice heisenberg models, Phys. Rev. Lett.99, 127004 (2007)

    2007

  56. [57]

    Kaneko, S

    R. Kaneko, S. Morita, and M. Imada, Gapless spin- liquid phase in an extended spin 1/2 triangular heisen- berg model, Journal of the Physical Society of Japan83, 093707 (2014)

    2014

  57. [58]

    Iqbal, W.-J

    Y. Iqbal, W.-J. Hu, R. Thomale, D. Poilblanc, and F. Becca, Spin liquid nature in the heisenbergJ1 −J 2 triangular antiferromagnet, Phys. Rev. B93, 144411 (2016)

    2016

  58. [59]

    K. G. Wilson and J. B. Kogut, The Renormalization group and the epsilon expansion, Phys. Rept.12, 75 (1974)

    1974

  59. [60]

    K. G. Wilson, The renormalization group: Critical phe- nomena and the kondo problem, Rev. Mod. Phys.47, 773 (1975)

    1975

  60. [61]

    Goldenfeld, Lectures on phase transitions and the renormalization group (1972)

    N. Goldenfeld, Lectures on phase transitions and the renormalization group (1972)

    1972

  61. [62]

    Cardy,Scaling and renormalization in statistical physics, Vol

    J. Cardy,Scaling and renormalization in statistical physics, Vol. 5 (Cambridge university press, 1996)

    1996

  62. [63]

    Nomura, Helping restricted boltzmann machines with quantum-state representation by restoring symmetry, JournalofPhysics: CondensedMatter33,174003(2021)

    Y. Nomura, Helping restricted boltzmann machines with quantum-state representation by restoring symmetry, JournalofPhysics: CondensedMatter33,174003(2021)

    2021

  63. [64]

    M. Reh, M. Schmitt, and M. Gärttner, Optimizing design choices for neural quantum states, Phys. Rev. B107, 195115 (2023)

    2023

  64. [65]

    M. B. Hastings, I. González, A. B. Kallin, and R. G. Melko, Measuring renyi entanglement entropy in quan- tum monte carlo simulations, Phys. Rev. Lett.104, 157201 (2010)

    2010

  65. [66]

    Passetti, D

    G. Passetti, D. Hofmann, P. Neitemeier, L. Grunwald, 10 M. A. Sentef, and D. M. Kennes, Can neural quantum states learn volume-law ground states?, Phys. Rev. Lett. 131, 036502 (2023)

    2023

  66. [67]

    can neural quantum states learn volume-law ground states?

    Z. Denis, A. Sinibaldi, and G. Carleo, Comment on “can neural quantum states learn volume-law ground states?”, Phys. Rev. Lett.134, 079701 (2025)

    2025

  67. [68]

    Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of Physics349, 117–158 (2014)

    R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Annals of Physics349, 117–158 (2014)

    2014

  68. [69]

    Turkeshi, E

    X. Turkeshi, E. Tirrito, and P. Sierant, Magic spread- ing inrandom quantum circuits, NatureCommunications 16, 2575 (2025)

    2025

  69. [70]

    P. S. Tarabunga, E. Tirrito, T. Chanda, and M. Dal- monte, Many-body magic via pauli-markov chains—from criticality to gauge theories, PRX Quantum4, 040317 (2023)

    2023

  70. [71]

    Leone, S

    L. Leone, S. F. E. Oliviero, and A. Hamma, Stabilizer rényi entropy, Phys. Rev. Lett.128, 050402 (2022)

    2022

  71. [72]

    Sinibaldi, A

    A. Sinibaldi, A. F. Mello, M. Collura, and G. Carleo, Nonstabilizerness of neural quantum states, Phys. Rev. Res.7, 043289 (2025)

    2025

  72. [73]

    Mendes-Santos, M

    T. Mendes-Santos, M. Schmitt, A. Angelone, A. Ro- driguez, P. Scholl, H. J. Williams, D. Barredo, T. La- haye, A. Browaeys, M. Heyl, and M. Dalmonte, Wave- function network description and kolmogorov complex- ity of quantum many-body systems, Phys. Rev. X14, 021029 (2024)

    2024

  73. [74]

    Zhang, Coexistence of superconductivity with par- tially filled stripes in the hubbard model, Science384, 10.1126/science.adh7691 (2024)

    H.Xu, C.-M.Chung, M.Qin, U.Schollwöck, S.R.White, and S. Zhang, Coexistence of superconductivity with par- tially filled stripes in the hubbard model, Science384, 10.1126/science.adh7691 (2024)

    work page doi:10.1126/science.adh7691 2024

  74. [75]

    C. Roth, A. Chen, A. Sengupta, and A. Georges, Superconductivity in the two-dimensional hubbard model revealed by neural quantum states (2025), arXiv:2511.07566 [cond-mat.supr-con]

    arXiv 2025

  75. [76]

    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)

    2020

  76. [77]

    J. Nys, G. Pescia, A. Sinibaldi, and G. Carleo, Ab-initio variational wave functions for the time-dependent many- electron schrödinger equation, Nature communications 15, 9404 (2024)

    2024

  77. [78]

    Rende, S

    R. Rende, S. Goldt, F. Becca, and L. L. Viteritti, Fine- tuning neural network quantum states, Phys. Rev. Res. 6, 043280 (2024)

    2024

  78. [79]

    Sorella, Green function monte carlo with stochastic reconfiguration, Phys

    S. Sorella, Green function monte carlo with stochastic reconfiguration, Phys. Rev. Lett.80, 4558 (1998)

    1998

  79. [80]

    Rende, L

    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, Communica- tions Physics7, 10.1038/s42005-024-01732-4 (2024)

    work page doi:10.1038/s42005-024-01732-4 2024

  80. [81]

    A.Sinibaldi, C.Giuliani, G.Carleo,andF.Vicentini,Un- biasing time-dependent variational monte carlo by pro- jected quantum evolution, Quantum7, 1131 (2023)

    2023

Showing first 80 references.