pith. sign in

arxiv: 2605.03043 · v1 · submitted 2026-05-04 · 🪐 quant-ph · cond-mat.dis-nn

Information in Many-body Eigenstates: A Question of Learnability

Maksymilian Kliczkowski , Jaros{\l}aw Paw{\l}owski , Masudul Haque This is my paper

Pith reviewed 2026-05-08 19:01 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.dis-nn
keywords many-body eigenstateslearnabilitymachine learningHamiltonian reconstructionspectral edgesentanglementneural networksquantum information
0
0 comments X

The pith

Eigenstates near the spectral edges of many-body Hamiltonians allow more accurate reconstruction of the Hamiltonian than mid-spectrum eigenstates when using an encoder-decoder neural network.

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

The paper defines learnability as the precision with which the Hamiltonian can be recovered from one or more eigenstates using machine learning. It contrasts low-entanglement, structured eigenstates at the spectrum edges with high-entanglement, near-random eigenstates in the middle. Using an encoder-decoder network and a physics-based loss, the authors show that edge eigenstates yield substantially higher prediction accuracy and require fewer samples for effective learning. This establishes learnability as an alternative, ML-enabled measure of how much information eigenstates carry about their parent Hamiltonian. A reader would care because the result supplies a concrete, operational distinction between the two classes of eigenstates that entanglement measures alone do not provide.

Core claim

For a fixed encoder-decoder architecture and physics-inspired loss, spectral-edge eigenstates permit markedly higher accuracy in Hamiltonian prediction and need fewer training examples than mid-spectrum eigenstates, quantifying the greater informational content of the former.

What carries the argument

Learnability, measured by the accuracy of Hamiltonian reconstruction from eigenstate input via an encoder-decoder neural network trained with a physics-inspired loss function.

If this is right

  • Spectral position determines the practical utility of eigenstates for Hamiltonian inference.
  • Fewer edge eigenstates suffice to learn the Hamiltonian compared with bulk eigenstates.
  • Machine learning supplies a quantitative probe of eigenstate information content distinct from entanglement entropy.
  • The distinction between edge and mid-spectrum eigenstates is manifested directly in their differential learnability.

Where Pith is reading between the lines

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

  • Experimental protocols that preferentially access low-energy eigenstates could reduce the resources needed for Hamiltonian learning in quantum simulators.
  • The same learnability test could be applied to disordered or driven systems to check whether localization alters the edge-bulk information gap.
  • Integrable versus chaotic models could be compared to test whether the learnability difference survives changes in level statistics.

Load-bearing premise

The encoder-decoder architecture and chosen loss function measure the intrinsic information content of the eigenstates rather than artifacts of the network or training procedure.

What would settle it

Achieving comparable Hamiltonian reconstruction accuracy from an equal number of mid-spectrum eigenstates as from edge eigenstates would falsify the claimed difference in learnability.

Figures

Figures reproduced from arXiv: 2605.03043 by Jaros{\l}aw Paw{\l}owski, Maksymilian Kliczkowski, Masudul Haque.

Figure 1
Figure 1. Figure 1: Typical distinction between eigenstates in differ view at source ↗
Figure 2
Figure 2. Figure 2: Sketch of the learnability protocol. (a) One-dimensional J1J2 spin-1/2 chain Hˆ (θ) [Eq. (1)], parametrized by a vector θ [Eq. (2)], is diagonalized to obtain a full set of eigenstates {|Ψ mi θ ⟩} and corresponding eigenvalues {E mi θ }. (b) Subsets of eigenstates are selected according to three spectral windows: (i) low-energy sector, (ii) middle of the spectrum, and (iii) single eigenstate mi. These stat… view at source ↗
Figure 4
Figure 4. Figure 4: Systematic dependence of Hamiltonian reconstruc view at source ↗
Figure 5
Figure 5. Figure 5: Dependence of the loss on the neural-network view at source ↗
Figure 5
Figure 5. Figure 5: As the width of the hidden layer wH is varied, cf. Sec. III A, reconstruction yields progressively better re￾sults. However, while increasing the network size speeds up convergence for edge states, it does not remedy the lack of information in the middle of the spectrum. This further confirms that failure in the bulk is governed by intrinsic properties of the eigenstates, rather than limi￾tations of model … view at source ↗
Figure 7
Figure 7. Figure 7: (a) displays the LRayleigh curve [Eq. (6)] used dur￾ing optimization, while view at source ↗
Figure 8
Figure 8. Figure 8: (a) Training history using the supervised loss view at source ↗
Figure 9
Figure 9. Figure 9: (a) Training history using N LRayleigh [Eq. (6)], evaluated in terms of (b) Lθ [Eq. (8)] for system size L = 6 and numbers of input eigenstates M = 2, 10. In all cases, wH = 128 and Nepo = 1000 and low protocol is used; cf. Sec. III A 2. The model simultaneously infers both couplings (J1, J2). As illustrated in view at source ↗
read the original abstract

To what extent do individual eigenstates encode information of their underlying Hamiltonian, and how does this depend on their spectral position? For many-body quantum systems, this issue is widely understood in terms of the differing nature of the eigenstates near the spectral edges (low-entanglement, highly-structured eigenstates) and those far from the spectral edges (high-entanglement, near-random eigenstates). Utilizing the availability of machine learning tools, we introduce a new way to quantify the information contained in eigenstates: for a particular learning architecture, how precisely can the Hamiltonian be reconstructed from a single eigenstate? We refer to this property as learnability; it serves as a new, alternative measure of the information content of eigenstates, made possible by machine learning. Using an encoder-decoder neural network and a physics-inspired loss function, we demonstrate how the distinction between two types of eigenstates is manifested as a difference in learnability. For spectral-edge eigenstates, the prediction accuracy is much better, and fewer eigenstates are required to learn the Hamiltonian, compared to mid-spectrum eigenstates.

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

1 major / 2 minor

Summary. The manuscript introduces 'learnability' as a machine-learning-based measure of the information about the underlying Hamiltonian encoded in individual many-body eigenstates. Using an encoder-decoder neural network trained with a physics-inspired loss, the authors report that eigenstates near the spectral edges permit substantially higher Hamiltonian reconstruction accuracy and require fewer samples for successful learning than mid-spectrum eigenstates, attributing the distinction to the lower entanglement and greater structure of edge states.

Significance. If the central distinction survives broader validation, the work supplies a new operational, data-driven proxy for eigenstate information content that complements entanglement-based diagnostics. It could inform studies of the eigenstate thermalization hypothesis, many-body localization, and practical Hamiltonian learning protocols from limited quantum data.

major comments (1)
  1. The interpretation that the reported learnability gap is intrinsic to eigenstate structure (rather than an artifact of the chosen encoder-decoder architecture and loss) is load-bearing for the claim that learnability constitutes a general new measure of information content. No ablations with alternative architectures (linear models, graph networks, or different inductive biases) or loss terms are presented to test whether the edge-versus-mid-spectrum distinction persists; because edge states are low-entanglement while mid-spectrum states obey volume-law entanglement, any architecture favoring local correlations could preferentially succeed on the former. This concern is not addressed by the single-architecture results.
minor comments (2)
  1. Quantitative details required for assessing soundness—dataset sizes, number of eigenstates per training set, error bars on reconstruction accuracy, and statistical significance tests—are absent from the abstract and not referenced in the provided description of the results.
  2. The manuscript does not discuss controls for network capacity, hyperparameter sensitivity, or training convergence that could affect the measured sample efficiency and accuracy differences.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive and detailed review of our manuscript. The concern regarding potential architecture dependence is well-taken, and we address it directly below while clarifying the scope of our claims.

read point-by-point responses

  1. Referee: The interpretation that the reported learnability gap is intrinsic to eigenstate structure (rather than an artifact of the chosen encoder-decoder architecture and loss) is load-bearing for the claim that learnability constitutes a general new measure of information content. No ablations with alternative architectures (linear models, graph networks, or different inductive biases) or loss terms are presented to test whether the edge-versus-mid-spectrum distinction persists; because edge states are low-entanglement while mid-spectrum states obey volume-law entanglement, any architecture favoring local correlations could preferentially succeed on the former. This concern is not addressed by the single-architecture results.

    Authors: We agree that the absence of architecture ablations leaves open the possibility that the observed learnability gap could be influenced by the specific inductive biases of the encoder-decoder network and physics-inspired loss. The manuscript explicitly qualifies learnability as a property 'for a particular learning architecture' (see abstract), and we attribute the gap to the intrinsic structural differences between edge and mid-spectrum eigenstates, particularly their entanglement scaling. To strengthen the interpretation, the revised manuscript will include a new subsection with results from a linear regression baseline and a minimal feedforward network, testing whether the edge-versus-mid-spectrum distinction in reconstruction accuracy persists. We will also expand the discussion to note the current limitation and the need for broader validation in future work. This addresses the core of the concern without claiming full generality at present. revision: partial

Circularity Check

0 steps flagged

No circularity: learnability is an empirical definition with observed outcomes

full rationale

The paper defines learnability directly as the reconstruction accuracy achieved by training an encoder-decoder network on eigenstates to recover the Hamiltonian parameters, using a physics-inspired loss. The central observation—that spectral-edge eigenstates yield higher accuracy and require fewer samples than mid-spectrum states—is reported as the empirical result of this training procedure rather than a quantity derived from prior fitted parameters or self-referential equations. No load-bearing self-citations, uniqueness theorems, or ansatzes are invoked to force the distinction; the result remains an independent measurement of the chosen architecture's performance on the two classes of states.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the premise that neural-network reconstruction performance faithfully reflects intrinsic Hamiltonian information in eigenstates; this is an unproven domain assumption rather than a derived result.

free parameters (1)
  • Encoder-decoder architecture and training hyperparameters
    Network depth, width, optimizer settings, and loss weighting are chosen to make the reconstruction task work but are not derived from first principles.
axioms (1)
  • domain assumption A physics-inspired loss function can serve as a faithful proxy for Hamiltonian reconstruction fidelity
    Invoked to justify the training objective; no independent justification is supplied in the abstract.

pith-pipeline@v0.9.0 · 5500 in / 1183 out tokens · 71309 ms · 2026-05-08T19:01:34.727080+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

126 extracted references · 126 canonical work pages · 1 internal anchor

  1. [1]

    2(a)], see Eq

    The encoder The procedure begins by performing exact diagonal- ization (ED) of a one-dimensional spin-1/2J 1J2 chain [Fig. 2(a)], see Eq. (1), for preselected parametersθ. Their choice is discussed later in Sec. IV. The encoder takes as input a matrix of many-body eigenstates,Ψθ ∈ RD×M, whereMdenotes the number of eigenstates in- cluded in a single realiz...

    work page

  2. [2]

    Spectral protocol We consider three selection protocols. In Sec. IVB, we use (i) low-energy states (blue),i.e., the first MHamiltonian eigenstates,m i ∈ {1, . . . , M}, and (ii) middle-spectrum states (red),i.e., a set ofM consecutive eigenstates centered around the eigen- state with indexm av, whose energy is closest to the mean energy,E θ mav ≈E av ≡Tr ...

    work page

  3. [3]

    Dependence of the loss on the selected eigenstate in- dexm i according to single state selection protocol

    The decoder The decoder maps latent variables˜θto a Hamiltonian ˆH˜θ = ˜F( ˜θ), whose eigenstatesΨ ˜θ are required to match 4 10□4 102Lrayleigh (a) /u1D441sam = 1000 /u1D464/u1D43B= 4 /u1D464/u1D43B= 64 /u1D464/u1D43B= 128 (b) /u1D441sam = 20000 0 10 20 30/u1D45A/u1D456 10□3 100L /u1D703 (c) /u1D441sam = 1000 0 10 20 30/u1D45A/u1D456 (d) /u1D441sam = 2000...

    work page 2024

  4. [4]

    Science , author =

    G. Carleo and M. Troyer, Science355, 602 (2017), https://www.science.org/doi/pdf/10.1126/science.aag2302

    work page doi:10.1126/science.aag2302 2017

  5. [5]

    Lange, A

    H. Lange, A. Van de Walle, A. Abedinnia, and A. Bohrdt, From Architectures to Applications: A Re- view of Neural Quantum States (2024)

    work page 2024

  6. [6]

    Neural- network quantum states for many-body physics,

    M. Medvidović and J. R. Moreno, The European Phys- ical Journal Plus139, 631 (2024), arXiv:2402.11014 [cond-mat]

    work page arXiv 2024

  7. [7]

    S. Dash, L. Gravina, F. Vicentini, M. Ferrero, and A. Georges, Communications Physics8, 10.1038/s42005-025-02005-4 (2025)

    work page doi:10.1038/s42005-025-02005-4 2025

  8. [8]

    Paul, Physical Review Letters136, 120403 (2026)

    N. Paul, Physical Review Letters136, 120403 (2026)

    work page 2026

  9. [9]

    Torlai, G

    G. Torlai, G. Mazzola, J. Carrasquilla, M. Troyer, R.Melko,andG.Carleo,NaturePhysics14,447(2018)

    work page 2018

  10. [10]

    Koutný, L

    D. Koutný, L. Motka, Z. c. v. Hradil, J. Řeháček, and L. L. Sánchez-Soto, Phys. Rev. A106, 012409 (2022)

    work page 2022

  11. [11]

    Baláž, M

    P. Baláž, M. Krawczyk, J. Pawłowski, and K. Roszak, Phys. Rev. A112, 022431 (2025)

    work page 2025

  12. [12]

    Krawczyk, P

    M. Krawczyk, P. Baláž, K. Roszak, and J. Pawłowski, Learning quantum tomography from incomplete mea- surements (2026), arXiv:2506.19428 [quant-ph]

    work page arXiv 2026

  13. [13]

    Broecker, J

    P. Broecker, J. Carrasquilla, R. G. Melko, and S. Trebst, Scientific Reports7, 8823 (2017)

    work page 2017

  14. [14]

    Carrasquilla and R

    J. Carrasquilla and R. G. Melko, Nature Physics13, 431 (2017)

    work page 2017

  15. [15]

    X.-Y. Dong, F. Pollmann, and X.-F. Zhang, Physical Review B99, 121104 (2019)

    work page 2019

  16. [16]

    B. S. Rem, N. Käming, M. Tarnowski, L. Asteria, N. Fläschner, C. Becker, K. Sengstock, and C. Weit- enberg, Nature Physics15, 917 (2019)

    work page 2019

  17. [17]

    Canabarro, F

    A. Canabarro, F. F. Fanchini, A. L. Malvezzi, R. Pereira, and R. Chaves, Physical Review B100, 045129 (2019)

    work page 2019

  18. [18]

    Shiina, H

    K. Shiina, H. Mori, Y. Okabe, and H. K. Lee, Scientific Reports10, 2177 (2020)

    work page 2020

  19. [19]

    Yu, L.-W

    Y. Yu, L.-W. Yu, W. Zhang, H. Zhang, X. Ouyang, Y. Liu, D.-L. Deng, and L. M. Duan, Experimental unsupervised learning of non-Hermitian knotted phases with solid-state spins (2021)

    work page 2021

  20. [20]

    Mahlow, F

    F. Mahlow, F. S. Luiz, A. L. Malvezzi, and F. F. Fan- chini, Scientific Reports13, 14411 (2023)

    work page 2023

  21. [21]

    G. S. Franco, F. Mahlow, P. M. Prado, G. E. L. Pexe, L. A. M. Rattighieri, and F. F. Fanchini, Quantum Phases Classification Using Quantum Machine Learn- 9 ing with SHAP-Driven Feature Selection (2025)

    work page 2025

  22. [22]

    Behler and M

    J. Behler and M. Parrinello, Physical Review Letters 98, 146401 (2007)

    work page 2007

  23. [23]

    Behler, The Journal of Chemical Physics145, 170901 (2016)

    J. Behler, The Journal of Chemical Physics145, 170901 (2016)

    work page 2016

  24. [24]

    F. A. Faber, A. Lindmaa, O. A. Von Lilienfeld, and R. Armiento, Physical Review Letters117, 135502 (2016)

    work page 2016

  25. [25]

    Artrith, A

    N. Artrith, A. Urban, and G. Ceder, Physical Review B 96, 014112 (2017)

    work page 2017

  26. [26]

    A. Seko, H. Hayashi, K. Nakayama, A. Takahashi, and I. Tanaka, Physical Review B95, 144110 (2017)

    work page 2017

  27. [27]

    Akinpelu, M

    A. Akinpelu, M. Bhullar, and Y. Yao, Journal of Physics: Condensed Matter36, 453001 (2024)

    work page 2024

  28. [28]

    Nematov and M

    D. Nematov and M. Hojamberdiev, Computational Condensed Matter45, e01139 (2025)

    work page 2025

  29. [29]

    Yoon, J.-H

    H. Yoon, J.-H. Sim, and M. J. Han, Physical Review B 98, 245101 (2018)

    work page 2018

  30. [30]

    Zhang, M

    R. Zhang, M. E. Merkel, S. Beck, and C. Ederer, Phys- ical Review Research4, 043082 (2022)

    work page 2022

  31. [31]

    Kliczkowski, L

    M. Kliczkowski, L. Keyes, S. Roy, T. Paiva, M. Rande- ria, N. Trivedi, and M. M. Maśka, Physical Review B 110, 115119 (2024)

    work page 2024

  32. [32]

    Vidal, J

    G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett.90, 227902 (2003)

    work page 2003

  33. [33]

    Calabrese and J

    P. Calabrese and J. Cardy, Journal of statistical me- chanics: theory and experiment2004, P06002 (2004)

    work page 2004

  34. [34]

    Kitaev and J

    A. Kitaev and J. Preskill, Phys. Rev. Lett.96, 110404 (2006)

    work page 2006

  35. [35]

    Levin and X.-G

    M. Levin and X.-G. Wen, Phys. Rev. Lett.96, 110405 (2006)

    work page 2006

  36. [36]

    Eisert, M

    J. Eisert, M. Cramer, and M. B. Plenio, Reviews of Modern Physics82, 277 (2010)

    work page 2010

  37. [37]

    Laflorencie, Physics Reports646, 1 (2016)

    N. Laflorencie, Physics Reports646, 1 (2016)

    work page 2016

  38. [38]

    Bianchi, L

    E. Bianchi, L. Hackl, M. Kieburg, M. Rigol, and L. Vid- mar, PRX Quantum3, 030201 (2022)

    work page 2022

  39. [39]

    D. J. Luitz, N. Laflorencie, and F. Alet, Journal of Statistical Mechanics: Theory and Experiment2014, P08007 (2014)

    work page 2014

  40. [40]

    Vermersch, M

    B. Vermersch, M. Ljubotina, J. I. Cirac, P. Zoller, M. Serbyn, and L. Piroli, Physical Review X14, 031035 (2024)

    work page 2024

  41. [41]

    E. H. Lieb and D. W. Robinson, Communications in Mathematical Physics28, 251 (1972)

    work page 1972

  42. [42]

    M. B. Hastings, Journal of Statistical Mechanics: The- ory and Experiment2007, P08024 (2007)

    work page 2007

  43. [43]

    Srednicki, Phys

    M. Srednicki, Phys. Rev. E50, 888 (1994)

    work page 1994

  44. [44]

    J. M. Deutsch, Phys. Rev. A43, 2046 (1991)

    work page 2046

  45. [45]

    Rigol, V

    M. Rigol, V. Dunjko, and M. Olshanii, Nature (London) 452, 854 (2008)

    work page 2008

  46. [46]

    D’Alessio, Y

    L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys.65, 239 (2016)

    work page 2016

  47. [47]

    J. R. Garrison and T. Grover, Physical Review X8, 021026 (2018)

    work page 2018

  48. [48]

    M.Haque, P.A.McClarty,andI.M.Khaymovich,Phys- ical Review E105, 014109 (2022)

    work page 2022

  49. [49]

    Huang, IEEE Journal on Selected Areas in Informa- tion Theory5, 694 (2024)

    Y. Huang, IEEE Journal on Selected Areas in Informa- tion Theory5, 694 (2024)

    work page 2024

  50. [50]

    Kliczkowski, R

    M. Kliczkowski, R. Świętek, L. Vidmar, and M. Rigol, Physical Review E107, 064119 (2023)

    work page 2023

  51. [51]

    C. M. Langlett and J. F. Rodriguez-Nieva, Physical Re- view Letters134, 230402 (2025)

    work page 2025

  52. [52]

    Patil, L

    R. Patil, L. Hackl, G. R. Fagan, and M. Rigol, Physical Review B108, 245101 (2023)

    work page 2023

  53. [53]

    J.F.Rodriguez-Nieva, C.Jonay,andV.Khemani,Phys- ical Review X14, 031014 (2024)

    work page 2024

  54. [54]

    Świ¸ etek, M

    R. Świ¸ etek, M. Kliczkowski, L. Vidmar, and M. Rigol, Physical Review E109, 024117 (2024)

    work page 2024

  55. [55]

    Y. Yauk, R. Patil, Y. Zhang, M. Rigol, and L. Hackl, Physical Review B110, 235154 (2024)

    work page 2024

  56. [56]

    Qi and D

    X.-L. Qi and D. Ranard, Quantum3, 159 (2019)

    work page 2019

  57. [57]

    D. Bank, N. Koenigstein, and R. Giryes, Autoencoders (2020)

    work page 2020

  58. [58]

    A. N. Gorban, V. A. Makarov, and I. Y. Tyukin, En- tropy22, 82 (2020)

    work page 2020

  59. [59]

    Pugliese, S

    R. Pugliese, S. Regondi, and R. Marini, Data Science and Management4, 19 (2021)

    work page 2021

  60. [60]

    Cybenko, Mathematics of Control, Signals, and Sys- tems2, 303 (1989)

    G. Cybenko, Mathematics of Control, Signals, and Sys- tems2, 303 (1989)

    work page 1989

  61. [61]

    Gallant, IEEE Transactions on Neural Networks1, 179 (1990)

    S. Gallant, IEEE Transactions on Neural Networks1, 179 (1990)

    work page 1990

  62. [62]

    Hornik, M

    K. Hornik, M. Stinchcombe, and H. White, Neural Net- works2, 359 (1989)

    work page 1989

  63. [63]

    440– 449

    H.T.SiegelmannandE.D.Sontag,inProceedings of the fifth annual workshop on Computational learning theory (ACM, Pittsburgh Pennsylvania USA, 1992) pp. 440– 449

    work page 1992

  64. [64]

    Vaswani, N

    A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, Attention Is All You Need (2017)

    work page 2017

  65. [65]

    Barron, IEEE Transactions on Information Theory 39, 930 (1993)

    A. Barron, IEEE Transactions on Information Theory 39, 930 (1993)

    work page 1993

  66. [66]

    X. Yuan, S. Endo, Q. Zhao, Y. Li, and S. C. Benjamin, Quantum3, 191 (2019)

    work page 2019

  67. [67]

    Cerezo, A

    M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Nature Reviews Physics3, 625 (2021)

    work page 2021

  68. [68]

    Raissi, P

    M. Raissi, P. Perdikaris, and G. Karniadakis, Journal of Computational Physics378, 686 (2019)

    work page 2019

  69. [69]

    Wang, and L

    G.E.Karniadakis, I.G.Kevrekidis, L.Lu, P.Perdikaris, S. Wang, and L. Yang, Nature Reviews Physics3, 422 (2021)

    work page 2021

  70. [71]

    D.-L. Deng, X. Li, and S. Das Sarma, Physical Review X7, 021021 (2017)

    work page 2017

  71. [72]

    Y. Chen, Y. Pan, G. Zhang, and S. Cheng, Quantum Science and Technology7, 015005 (2021)

    work page 2021

  72. [73]

    Koutný, L

    D. Koutný, L. Ginés, M. Moczała-Dusanowska, S. Höfling, C. Schneider, A. Predojević, and M. Ježek, Science Advances9, eadd7131 (2023)

    work page 2023

  73. [74]

    Pawłowski and M

    J. Pawłowski and M. Krawczyk, Phys. Rev. Appl.22, 014068 (2024)

    work page 2024

  74. [75]

    Feng and L

    C. Feng and L. Chen, Communications in Theoretical Physics76, 075104 (2024)

    work page 2024

  75. [76]

    Huang, L

    Y. Huang, L. Che, C. Wei, F. Xu, X. Nie, J. Li, D. Lu, and T. Xin, npj Quantum Information11, 29 (2025)

    work page 2025

  76. [77]

    Krawczyk, J

    M. Krawczyk, J. Pawłowski, M. M. Maśka, and K. Roszak, Phys. Rev. A109, 022405 (2024)

    work page 2024

  77. [78]

    Zhang, P

    Y. Zhang, P. Ginsparg, and E.-A. Kim, Physical Review Research2, 023283 (2020)

    work page 2020

  78. [79]

    Valenti, G

    A. Valenti, G. Jin, J. Léonard, S. D. Huber, and E. Gre- plova, Physical Review A105, 023302 (2022)

    work page 2022

  79. [80]

    S. J. Wetzel, S. Ha, R. Iten, M. Klopotek, and Z. Liu, Interpretable Machine Learning in Physics: A Review (2025)

    work page 2025

  80. [81]

    E. P. L. Van Nieuwenburg, Y.-H. Liu, and S. D. Huber, 10 Nature Physics13, 435 (2017)

    work page 2017

Showing first 80 references.