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REVIEW 1 major objections 5 minor 56 references

Exact, training-free neural networks can represent Motzkin spin-chain states whose entanglement scales as log N or sqrt N, with parameter counts from O(1) to O(N^3).

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2026-08-01 04:25 UTC pith:HCUOSYFT

load-bearing objection Useful exact NQS constructions for Motzkin states, but the colorful transformer's exactness needs an explicit ω→∞/hard-attention limit — fixable, not fatal. the 1 major comments →

arxiv 2607.22522 v1 pith:HCUOSYFT submitted 2026-07-24 cond-mat.str-el quant-ph

Exact Neural-Network Representations of the Motzkin States

Runde Zha , Yuntian Gu , Chaohui Fan , Jia-lin Chen , Hai-Jun Liao , Tao Xiang This is my paper
classification cond-mat.str-el quant-ph PACS 03.65.Ud05.30.-d
keywords Motzkin spin chainneural quantum statesexact representationentanglementprefix sumtransformerrecurrent neural networktensor network
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper constructs exact neural-network wavefunctions for the colorless and colorful Motzkin spin chains, two frustration-free models whose ground states violate the entanglement area law. The weights are fixed analytically, so no training is needed; instead, the network acts as a legality checker that reads a spin configuration, verifies the Motzkin height constraint, and (for the colorful case) the last-in-first-out color-matching rule, and outputs 1 for exactly the valid configurations. The authors show this is possible in four mainstream architectures—recurrent, feedforward, convolutional, and transformer—with parameter scalings that match or beat known exact tensor-network representations. If correct, this demonstrates that neural quantum states can exactly capture highly nonlocal, area-law-violating entanglement, and provides a constructive template for designing neural-network ansatze for other constrained quantum systems.

Core claim

The central claim is that the support of the colorless and colorful Motzkin states can be encoded exactly by finite, fixed-weight neural networks. The colorless construction computes every prefix height through a causal prefix-sum module—implementable as a recurrent update, a dense triangular map, a one-sided convolution, or a masked attention layer—and then applies position-selective ReLU gates that kill any configuration whose height goes negative or whose final height is nonzero. The colorful construction adds a stack or attention-pointer module that tracks the most recent unmatched up step and enforces that each down step carries the same color. The resulting wavefunction is 1 on exactly

What carries the argument

The core mechanism is a causal prefix-sum module combined with position-selective rectified linear gates, which converts the Motzkin height constraints (nonnegativity and return to zero) into a product of gates that are 1 exactly for legal paths. For the colorful states, this is augmented by a causal stack module—or equivalently an attention pointer to the most recent unmatched up step—that enforces the last-in-first-out color-matching rule. The parameter count scales with system size N as O(1) to O(N^3) depending on the architecture, compared with O(N log^2 N) and O(s^4 N^2) for the corresponding exact tensor networks.

Load-bearing premise

For the colorful transformer, the attention layer is assumed to output exactly the color of the matching up step, but this holds only in the ω→∞ (or hard-mask) limit; for any finite ω the softmax mixes non-matching sites, so the color-violation flag can misfire on valid paths.

What would settle it

Evaluate the colorful transformer's wavefunction on all length-N colorful Motzkin paths with a large but finite ω (e.g., ω=10^6) and check whether Ψ(X)=1 for every valid path and 0 for every invalid path. Any valid path with a color-violation flag Γ_t=1, or any invalid path with Γ_t=0 for all t, would disprove exactness for finite ω.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Neural quantum states can exactly represent ground states with logarithmic and sqrt(N) entanglement, going beyond the area-law limitation that constrains matrix product states.
  • The fixed-weight, training-free constructions provide exact benchmarks for studying optimization, sampling, and generalization in neural quantum states.
  • The design principle—compile a causal legality algorithm into a neural network—extends naturally to other constrained states, such as Dyck/Fredkin paths, gauge-theory wavefunctions, stabilizer states, and string-net or loop-gas states.
  • The transformer construction is differentiable and can serve as a variational neural quantum state for perturbed Motzkin models that are not exactly solvable.
  • Parameter counts for the neural constructions are competitive with or lower than existing exact tensor-network representations for the colorful case.

Where Pith is reading between the lines

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

  • The exactness of the colorful transformer hinges on the ω→∞ limit; for finite ω the softmax attention weights are not exactly one-hot, so the attention output A2_t is only approximately the matching up-step color. A hard mask or an explicit pointer would remove this approximation entirely.
  • Because the paper measures parameter counts for fixed analytic weights, the comparison with tensor networks is about ansatz expressiveness in the zero-training-error sense, not about practical optimization cost; a trained network may need many more effective parameters to reach the same accuracy.
  • The recognition-computation perspective suggests that any quantum state whose support is decidable by a polynomial-size causal algorithm may admit an exact neural representation with similar complexity—this is the authors' stated design principle, but the paper only demonstrates it for Motzkin states.
  • A direct testable extension would be to apply the same prefix-sum-plus-stack compilation to the Fredkin or area-deformed Motzkin chains, whose ground states share the height and color-matching structure but with different amplitude weights.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper constructs fixed-weight, training-free neural-network representations of the colorless and colorful Motzkin states. For the colorless case it gives RNN, FNN, CNN, and transformer circuits that compute prefix heights and enforce nonnegativity and return-to-zero via ReLU gates. For the colorful case it augments the height module with a last-in-first-out color stack (RNN), a nearest-up-step pointer (FNN/CNN), or an attention pointer (transformer), and claims exact uniform amplitudes on the legal support. Parameter counts are tabulated and compared with tensor-network constructions, and verification code is provided.

Significance. If the claims hold, the paper provides a constructive, exact benchmark family for neural quantum states on states with log and sqrt(N) entanglement, showing that compact expressive representations can coexist with area-law violation. The colorless constructions and the colorful RNN/FNN/CNN constructions are mathematically clean and are genuine compilations of the path-validity rules rather than fitted ansatze; the code and fixed-weight design are valuable assets. The main weakness is the colorful transformer, whose finite-ω softmax pointer is not exact as written, so the four-architecture exactness claim is not yet fully established.

major comments (1)
  1. [Sec. IV C, Eqs. (79)-(84), (90)] The equality A2_t = x_j in Eq. (83) is not implied by the construction as written. The attention weights are softmax weights, so for any finite ω the output is a convex combination of site colors; the matching up step receives weight strictly less than 1 and non-matching sites receive nonzero weight. Consequently Γ_t in Eq. (84) can fire on valid colorful paths and Ψ(X)=σ(y_N+v_N−1) is not identically 1 on the legal support. The text only states ω≫1, which is an approximation statement, not an exactness statement. This is load-bearing because the abstract and introduction claim exact representations across all four architectures. Please specify hard/argmax attention explicitly, or take an explicit ω→∞ limit for fixed N and prove convergence of the softmax pointer to the matching up step. Also specify the behavior when no prior up step exists; the current 'nearly uniform' wording is only
minor comments (5)
  1. [Sec. IV C, text before Eq. (69)] The phrase 'hard softmax attention' is ambiguous and appears inconsistent with the softmax formulas in Eqs. (74) and (82). If hard attention is intended, please define it and replace the softmax weight formulas; if softmax is intended, the exactness caveat of the major comment applies.
  2. [Sec. IV C, Eq. (84)] Please specify the numerical encoding of colorful configurations in the transformer input x_t (for example +k for u_k, -k for d_k, 0 for 0) so that the condition x_t + A2_t = 0 in Γ_t is unambiguous. The earlier sections treat x_i as a height increment or a basis label, and the transformer uses x_t as a scalar.
  3. [Sec. IV C, opening paragraph] The construction is described as a 'multi-head transformer,' but the attention layers appear to use a single attention head each. Either remove 'multi-head' or specify the head structure.
  4. [Sec. IV A, Eq. (48)] The notation q^{S_{i-1}}_{i-1} for the stack top is undefined when S_{i-1}=0. Please define the sentinel value used in that case so that the pop rule and Eq. (49) are well-defined for a down step from height zero.
  5. [Table I and Sec. V A] The colorless transformer is listed as O(N^2) because of the (N+2)-dimensional embedding, while the colorful transformer is listed as O(1) because of the fixed 9-dimensional construction. Please state explicitly that the O(1) count assumes the fixed masks and, if the finite-ω issue is resolved by a hard-attention limit, clarify whether ω is counted as a parameter.

Circularity Check

0 steps flagged

No significant circularity: the constructions compile the Motzkin legality rules into fixed-weight networks, with no fitted parameter or self-citation used as the load-bearing premise.

full rationale

The derivation chain starts from the Motzkin path constraints (Eqs. 1-5) and the parent-Hamiltonian results [8,9], then constructs each network by explicitly computing prefix sums and legality gates. The networks' weights and masks are fixed analytically: lower-triangular W (Eq. 26), causal mask (Eqs. 34-37), ReLU gates (Eqs. 42-44), stack updates (Eqs. 48-49), and projection matrices (Eqs. 70-81). No parameter is fitted to the target wavefunction, and no 'prediction' is obtained from a quantity that was itself defined in terms of that prediction. The claims like Ψ(X)=1 iff X is a valid Motzkin path are direct identities over integer-valued features, not circular reductions. Self-citations [8,9,12,13] supply background results or comparison baselines, but the central network legality argument does not rest on them. The most notable manuscript gap is the colorful transformer's assertion that A2_t = x_j for finite but large ω (Eq. 83): this is an approximation/unproven-limit issue, not an input-output equivalence by construction, so it belongs to correctness risk rather than circularity. The paper also provides external verification code [54], further supporting that the constructions are self-contained. No circular step of any of the enumerated kinds was found.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The construction introduces one hyperparameter ω; no physical entities are postulated. The paper relies on known Motzkin-state definitions and on standard facts about ReLU and attention.

free parameters (1)
  • ω (colorful transformer attention sharpness) = arbitrary large (ω ≫ 1)
    Controls softmax selection of the matching up step in Layer 2; exactness requires ω→∞, but for finite ω the attention output is approximate. Introduced in Sec. IV C.
axioms (5)
  • domain assumption Definitions and ground-state properties of colorless and colorful Motzkin states from Refs. [8,9] (equal-weight superpositions over Motzkin paths).
    The paper builds representations of these known states; it does not re-derive them.
  • standard math ReLU networks can exactly implement Kronecker deltas via σ(1-|x-y|) for integer arguments.
    Used in the FNN activation, Eqs. (60)-(62).
  • standard math Softmax with -∞ logits yields exact hard causal attention.
    Used in the colorless transformer, Eqs. (36)-(37).
  • domain assumption The most recent preceding up step at the same height is the LIFO matching partner for a down step.
    Underpins the pointer construction in the colorful FNN and transformer; not proven in the paper but standard for Dyck paths.
  • domain assumption The stack update rules in the colorful RNN correctly track unmatched colors.
    Stated in Sec. IV, Eqs. (48)-(49).

pith-pipeline@v1.3.0-alltime-deepseek · 15930 in / 22169 out tokens · 208403 ms · 2026-08-01T04:25:57.086042+00:00 · methodology

0 comments
read the original abstract

Motzkin spin chains are paradigmatic frustration-free one-dimensional quantum systems whose ground states feature exactly solvable combinatorial structures and exotic, area-law-violating entanglement scaling. Specifically, colorless Motzkin states exhibit critical logarithmic entanglement divergence \(\log N\) with system size \(N\), while their colorful counterparts host supercritical sublinear \(\sqrt{N}\) entanglement growth. Such unconventional entanglement behaviors place these states well beyond the expressive capability of standard matrix product states, which are fundamentally constrained by the entanglement area law. Here, we systematically construct exact, training-free neural-network representations for both colorless and colorful Motzkin states across four mainstream architectures, including recurrent, feedforward, convolutional, and transformer networks. Our core design leverages a causal prefix-sum module, implementable via recurrent updates, feedforward mappings, or masked attention layers, combined with position-selective rectified linear gates that enforce the Motzkin height constraints. For the colorful states, we further introduce a dedicated causal stack module that explicitly encodes the last-in-first-out color-matching rule. Our results demonstrate that neural architectures can accurately capture highly non-trivial entanglement features inaccessible to conventional tensor networks, providing prototypic examples for benchmarking and a constructive design framework for future neural-network quantum state developments targeting strongly entangled quantum systems.

Figures

Figures reproduced from arXiv: 2607.22522 by Chaohui Fan, Hai-Jun Liao, Jia-lin Chen, Runde Zha, Tao Xiang, Yuntian Gu.

Figure 1
Figure 1. Figure 1: FIG. 1. Examples of Motzkin-path constraints. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

56 extracted references · 5 linked inside Pith

  1. [1]

    (ei)j =δ i,j,(33) and the last component is left empty to receive the at- tention output

    Attention block In this transformer construction, each site is first em- bedded as z0 i = (xi, ei,0) T ∈R N+2 ,(32) wheree i is theith standard basis vector in position space, i.e. (ei)j =δ i,j,(33) and the last component is left empty to receive the at- tention output. The attention block uses the projection matrices WQ, WK, WV to map the inputz 0 into q...

  2. [2]

    (44) SinceS t is integer-valued,ψ t = 1 iff the corresponding Motzkin height constraint is satisfied, whereasψ t = 0 otherwise

    Activation Starting fromz 1 t = (xt, et, ¯St)T , obtained from the at- tention block with a residual connection, we define the hidden unit at sitetas ht(z1 t ) =σ −t ¯St +σ(1−N+t)σ t ¯St =σ(−S t) +δ t,N σ(S t).(42) The corresponding site gate is ψt =σ 1−h t(z1 t ) ,(43) which can be written explicitly as ψt = ( σ[1−σ(−S t)], t < N, σ[1−σ(−S N )−σ(S N )], ...

  3. [3]

    M. B. Hastings, An area law for one-dimensional quan- tum systems, J. Stat. Mech.2007, P08024 (2007)

    2007

  4. [4]

    Eisert, M

    J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010)

    2010

  5. [5]

    Holzhey, F

    C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B424, 443 (1994)

    1994

  6. [6]

    Vidal, J

    G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entan- glement in quantum critical phenomena, Phys. Rev. Lett. 90, 227902 (2003)

    2003

  7. [7]

    V. E. Korepin, Universality of entropy scaling in one di- mensional gapless models, Phys. Rev. Lett.92, 096402 (2004). 13

    2004

  8. [8]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech.2004, P06002 (2004)

    2004

  9. [9]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A42, 504005 (2009)

    2009

  10. [10]

    Bravyi, L

    S. Bravyi, L. Caha, R. Movassagh, D. Nagaj, and P. W. Shor, Criticality without frustration for quantum spin-1 chains, Phys. Rev. Lett.109, 207202 (2012)

    2012

  11. [11]

    Movassagh and P

    R. Movassagh and P. W. Shor, Supercritical entangle- ment in local systems: Counterexample to the area law for quantum matter, Proc. Natl. Acad. Sci.113, 13278 (2016)

    2016

  12. [12]

    Feldmeier, Y.-J

    J. Feldmeier, Y.-J. Liu, M. D. Lukin, and S. Choi, Digital dissipative state preparation for frustration-free gapless quantum systems (2026), arXiv:2603.10119

    arXiv 2026

  13. [13]

    Mukherjee, H

    K. Mukherjee, H. Barghathi, A. Del Maestro, and R. Mukherjee, Quantum simulation of Motzkin spin chain with Rydberg atoms (2026), arXiv:2603.23422

    arXiv 2026

  14. [14]

    R. N. Alexander, G. Evenbly, and I. Klich, Exact holo- graphic tensor networks for the Motzkin spin chain, Quantum5, 546 (2021)

    2021

  15. [15]

    R. N. Alexander, A. Ahmadain, Z. Zhang, and I. Klich, Exact rainbow tensor networks for the colorful Motzkin and Fredkin spin chains, Phys. Rev. B100, 214430 (2019)

    2019

  16. [16]

    Carleo and M

    G. Carleo and M. Troyer, Solving the quantum many- body problem with artificial neural networks, Science 355, 602 (2017)

    2017

  17. [17]

    Gao and L.-M

    X. Gao and L.-M. Duan, Efficient representation of quan- tum many-body states with deep neural networks, Nat. Commun.8, 662 (2017)

    2017

  18. [18]

    J. Chen, S. Cheng, H. Xie, L. Wang, and T. Xiang, Equiv- alence of restricted Boltzmann machines and tensor net- work states, Phys. Rev. B97, 085104 (2018)

    2018

  19. [19]

    Huang and J

    Y. Huang and J. E. Moore, Neural network representa- tion of tensor network and chiral states, Phys. Rev. Lett. 127, 170601 (2021)

    2021

  20. [20]

    Z.-Y. Han, J. Wang, H. Fan, L. Wang, and P. Zhang, Unsupervised generative modeling using matrix product states, Phys. Rev. X8, 031012 (2018)

    2018

  21. [21]

    Cheng, L

    S. Cheng, L. Wang, T. Xiang, and P. Zhang, Tree ten- sor networks for generative modeling, Phys. Rev. B99, 155131 (2019)

    2019

  22. [22]

    S. Li, F. Pan, P. Zhou, and P. Zhang, Boltzmann ma- chines as two-dimensional tensor networks, Phys. Rev. B 104, 075154 (2021)

    2021

  23. [23]

    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

    arXiv 2026

  24. [24]

    K. Choo, T. Neupert, and G. Carleo, Two-dimensional frustratedJ 1 −J 2 model studied with neural network quantum states, Phys. Rev. B100, 125124 (2019)

    2019

  25. [25]

    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

  26. [26]

    Nomura, A

    Y. Nomura, A. S. Darmawan, Y. Yamaji, and M. Imada, Restricted Boltzmann machine learning for solving strongly correlated quantum systems, Phys. Rev. B96, 205152 (2017)

    2017

  27. [27]

    Sharir, Y

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

    2020

  28. [28]

    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

  29. [29]

    Gauvin-Ndiaye, J

    C. Gauvin-Ndiaye, J. Tindall, J. R. Moreno, and A. Georges, Mott transition and volume law en- tanglement with neural quantum states (2024), arXiv:2311.05749

    Pith/arXiv arXiv 2024

  30. [30]

    Lange, A

    H. Lange, A. B¨ ohler, C. Roth, and A. Bohrdt, Simulating the two-dimensionalt−Jmodel at finite doping with neu- ral quantum states, Phys. Rev. Lett.135, 136504 (2025)

    2025

  31. [31]

    Chen, Z.-Q

    A. Chen, Z.-Q. Wan, A. Sengupta, A. Georges, and C. Roth, Neural network-augmented Pfaffian wave- functions for scalable simulations of interacting fermions (2025), arXiv:2507.10705

    Pith/arXiv arXiv 2025

  32. [32]

    K. Choo, A. Mezzacapo, and G. Carleo, Fermionic neural-network states for ab-initio electronic structure, Nat. Commun.11, 2368 (2020)

    2020

  33. [33]

    Robledo Moreno, G

    J. Robledo Moreno, G. Carleo, A. Georges, and J. Stokes, Fermionic wave functions from neural-network con- strained hidden states, Proc. Natl. Acad. Sci.119, e2122059119 (2022)

    2022

  34. [34]

    Liu and B

    Z.-C. Liu and B. K. Clark, Unifying view of fermionic neural network quantum states: From neural network backflow to hidden fermion determinant states, Phys. Rev. B110, 115124 (2024)

    2024

  35. [35]

    A. Chen, C. Roth, Z.-Q. Wan, A. Sengupta, and A. Georges, Scalable and accurate simulations of the hubbard model with neural quantum states (2025), pre- sented at the Machine Learning and the Physical Sciences (ML4PS) Workshop, NeurIPS 2025

    2025

  36. [36]

    Ibarra-Garc ´ ıa-Padilla, H

    E. Ibarra-Garc ´ ıa-Padilla, H. Lange, R. G. Melko, R. T. Scalettar, J. Carrasquilla, A. Bohrdt, and E. Khatami, Autoregressive neural quantum states of Fermi Hubbard models, Phys. Rev. Res.7, 013122 (2025)

    2025

  37. [37]

    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. (2026)

    2026

  38. [38]

    Y. Gu, Z. Han, W. Li, Z. Xiao, T. Xiang, M. Qin, L. Wang, and D. Lv, Pareto frontier of neural quantum states: Scalable, affordable, and accurate convolutional backflow for strongly correlated lattice fermions (2026), arXiv:2604.25775

    Pith/arXiv arXiv 2026

  39. [39]

    Zhouyin, T.-H

    Z. Zhouyin, T.-H. Lee, A. Chen, N. Lanat` a, and H. Guo, Neural-quantum-states impurity solver for quantum em- bedding problems, Phys. Rev. B113, 155123 (2026)

    2026

  40. [40]

    J. L. Elman, Finding structure in time, Cognitive Science 14, 179 (1990)

    1990

  41. [41]

    Hochreiter and J

    S. Hochreiter and J. Schmidhuber, Long short-term mem- ory, Neural Comput.9, 1735 (1997)

    1997

  42. [42]

    Rosenblatt, The perceptron: A probabilistic model for information storage and organization in the brain, Psychological Review65, 386 (1958)

    F. Rosenblatt, The perceptron: A probabilistic model for information storage and organization in the brain, Psychological Review65, 386 (1958)

    1958

  43. [43]

    D. E. Rumelhart, G. E. Hinton, and R. J. Williams, Learning representations by back-propagating errors, Na- ture323, 533 (1986)

    1986

  44. [44]

    Cybenko, Approximation by superpositions of a sig- moidal function, Math

    G. Cybenko, Approximation by superpositions of a sig- moidal function, Math. Control Signals Syst.2, 303 (1989)

    1989

  45. [45]

    LeCun, L

    Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, Gradient-based learning applied to document recogni- tion, Proc. IEEE86, 2278 (1998)

    1998

  46. [46]

    Krizhevsky, I

    A. Krizhevsky, I. Sutskever, and G. E. Hinton, ImageNet classification with deep convolutional neural networks, 14 Advances in Neural Information Processing Systems 25 , 1097 (2012)

    2012

  47. [47]

    K. He, X. Zhang, S. Ren, and J. Sun, Deep residual learn- ing for image recognition, Proc. IEEE Conf. Comput. Vis. Pattern Recognit. , 770 (2016)

    2016

  48. [48]

    Liang, W.-Y

    X. Liang, W.-Y. Liu, P.-Z. Lin, G.-C. Guo, Y.-S. Zhang, and L. He, Solving frustrated quantum many-particle models with convolutional neural networks, Phys. Rev. B98, 104426 (2018)

    2018

  49. [49]

    Vaswani, N

    A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, At- tention is all you need, Advances in Neural Information Processing Systems 30 , 5998 (2017), arXiv:1706.03762

    Pith/arXiv arXiv 2017

  50. [50]

    Devlin, M.-W

    J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, BERT: Pre-training of deep bidirectional transformers for language understanding, Proc. NAACL-HLT , 4171 (2019)

    2019

  51. [51]

    Zhang and M

    Y.-H. Zhang and M. Di Ventra, Transformer quantum state: A multipurpose model for quantum many-body problems, Phys. Rev. B107, 075147 (2023)

    2023

  52. [52]

    Menon, A

    V. Menon, A. Gu, and R. Movassagh, Symmetries, cor- relation functions, and entanglement of general quantum Motzkin spin-chains (2024), arXiv:2408.16070

    Pith/arXiv arXiv 2024

  53. [53]

    Zhang, A

    Z. Zhang, A. Ahmadain, and I. Klich, Novel quantum phase transition from bounded to extensive entangle- ment, Proc. Natl. Acad. Sci.114(20), 5142 (2017)

    2017

  54. [54]

    Levine and R

    L. Levine and R. Movassagh, The gap of the area- weighted Motzkin spin chain is exponentially small, J. Phys. A50, 255302 (2017)

    2017

  55. [55]

    Salberger and V

    O. Salberger and V. E. Korepin, Entangled spin chain, Rev. Math. Phys.29(2017)

    2017

  56. [56]

    Verification code for exact neural-network represen- tations of the motzkin states,https://github.com/ ArtistET/Neuralnetwork_representation_motzkin (2026)

    2026