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REVIEW 4 major objections 6 minor 64 references

Probing Quantum Spin Systems with Kolmogorov-Arnold Neural Network Quantum States

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read SineKAN neural networks can represent quantum spin ground states more cheaply and accurately than RBM, LSTM, and MLP baselines.

desk verdict Useful and honest KAN-NQS benchmark, but the L=100 superiority claim needs seed-averaged error bars before I'd trust it. read the letter →

arxiv 2506.01891 v4 pith:L2RIHKXN submitted 2025-06-02 quant-ph cond-mat.dis-nncond-mat.str-elcs.LG

Mahmud Ashraf Shamim , Eric A F Reinhardt , Talal Ahmed Chowdhury , Sergei Gleyzer , Paulo T Araujo This is my paper
classification quant-phcond-mat.dis-nncond-mat.str-elcs.LG
keywords SineKANKolmogorov-ArnoldnetworksneuralquantumstatesvariationalMonteCarloJ1-J2modelMarshallsignrulespinchainsreflectionsymmetry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

SineKAN is proposed as a neural quantum state ansatz that represents a many-body wavefunction as a nested sum of learnable univariate sinusoidal functions, following the Kolmogorov-Arnold superposition theorem. The paper shows that the plain and reflection-symmetric variants match exact diagonalization for 20-site transverse-field Ising and anisotropic Heisenberg chains, and reproduce density-matrix renormalization group ground-state energies, fidelities, and correlation functions for the frustrated $J_1$-$J_2$ chain at sizes up to 100 sites. For the 100-site $J_1$-$J_2$ model, the symmetric rSineKAN achieves lower energy error and variance than restricted Boltzmann machines, LSTMs, and MLPs with comparable or larger parameter counts. The work argues that KAN-style learnable on-edge activations are a computationally cheap and expressive alternative for variational quantum many-body calculations.

What carries the argument

The load-bearing object is the SineKAN layer, a sinusoidal activation with learnable frequency $\omega_n$ and amplitude $A_{mnl}$ placed on every edge of the network, with a fixed grid of phase shifts $\pi n/N + \pi l/L$ plus a small random perturbation $\delta_{nl}$. Stacking such layers produces the nested univariate function of Eq. (2), whose parameters are optimized with variational Monte Carlo to minimize the Hamiltonian expectation value. Reflection symmetry enters through $y = S(\vec{\sigma}) + S(R(\vec{\sigma}))$, sharing weights between the chain and its mirror image and halving the effective parameter space. The Marshall sign rule is applied at the Hamiltonian level rather than in the network's phase, which is what lets the real, sign-free ansatz represent the alternating-sign ground states of the bipartite and weakly frustrated regimes.

What would settle it

Take the 100-site $J_1$-$J_2$ chain at $J_2=0.4$ or $0.5$, train rSineKAN both with and without the Marshall-sign-rule rotation applied to the Hamiltonian, and compare each variational energy and fidelity to DMRG; if the gap between the two runs is large, the reported advantage over RBM, LSTM, and MLP is an artifact of the sign encoding rather than of SineKAN expressivity. A second check is to rerun the $L=64$ case with grid size 8, where Table II shows the energy collapsing to $-16.962$ against an exact $-24$; if the collapse reappears under different random seeds, the anomaly is a genuine resonance, not a tuning accident.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that a SineKAN wavefunction, built from layers of the form $y_m = \sum_{l,n} A_{mnl}\sin(\omega_n x_l + \pi n/N + \pi l/L + \delta_{nl}) + b_m$ with learnable amplitudes, frequencies, and biases, can serve as a highly accurate neural quantum state. In the $J_1$-$J_2$ chain with $L=100$ sites, the reflection-symmetric variant rSineKAN, constructed by summing the network's output on a spin configuration and its center reflection, outperforms RBM, LSTM, and MLP ans\"atze in reproducing DMRG ground-state energies while using about 86,000 parameters. The ansatz is trained by variational Monte Carlo with the Marshall sign rule encoded into the Hamiltonian, and its accuracy holds across the gapless to dimerized transition up to the Majumdar-Ghosh point at $J_2=0.5$; at $J_2=0.6$, where the sign rule breaks down, performance degrades for every ansatz tested. The paper also documents a sharp failure when the hidden-layer size matches the chain length and the grid size equals $\sqrt{L}$, avoided by perturbing the grid size by one.

Load-bearing premise

The load-bearing premise is that the Marshall sign rule, which the paper encodes directly into the $J_1$-$J_2$ Hamiltonian for every value of $J_2$, is a faithful description of the true ground-state sign structure wherever the paper's accuracy claims are made; the rule is exact only at $J_2=0$ and only approximate beyond it, so a sign-structure mismatch would bias the reported energies and fidelities.

Editorial extensions

If this is right

  • For 1D spin chains, a neural quantum state can be made markedly more parameter-efficient by replacing fixed node activations with learnable on-edge sine functions; the 100-site $J_1$-$J_2$ runs use tens of thousands of parameters rather than the millions of an RBM baseline.
  • The reflection-symmetric rSineKAN resolves the near-degenerate ground states of the anisotropic Heisenberg chain and the Majumdar-Ghosh point, indicating that simple discrete symmetries can substitute for much larger model capacity.
  • SineKAN inference time scales almost flatly with chain length up to hundreds of sites, so the ansatz is practical for systems where RBM inference has already become exponentially costly.
  • Beyond $J_2=0.5$, the fidelity drop tracks the violation of the Marshall sign rule, turning the ansatz into a tool for diagnosing where sign structure is no longer captured by a fixed phase convention.
  • If the comparison holds, transformer-free NQS architectures based on KAN layers are competitive baselines for future frustrated-magnetism studies, including the previously studied $L=100$ chain.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The grid-size failure at $\sqrt{L}$ suggests a periodicity collision between the sinusoidal grid phases and the input lattice; a testable remedy is to use incommensurate phase shifts or random grids, which would make the anomaly vanish without hand tuning.
  • The paper's comparison encodes the Marshall sign rule for all ans\"atze, so the cleanest extension is to test rSineKAN on genuinely non-bipartite or geometrically frustrated models, where no useful sign rule exists and the network must learn signs from scratch.
  • If the computational savings replicate, SineKAN layers could replace MLP heads inside transformer or autoregressive NQS architectures, giving those models a cheaper way to represent local correlations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The manuscript proposes SineKAN, a neural quantum state (NQS) ansatz built from Kolmogorov-Arnold networks with learnable sinusoidal activations, and benchmarks it on three one-dimensional spin models. For L=20, the vSineKAN and rSineKAN variants are compared against exact diagonalization for the transverse-field Ising model, the anisotropic Heisenberg model, and the J1-J2 model, including energies, variances, fidelities, correlations, and structure factors. For L=32, 64, and 100, ground-state energies are compared with DMRG, and at L=100 the authors compare against RBM, RNN-LSTM, and MLP baselines. The central claim is that the reflection-symmetric rSineKAN outperforms these baselines in energy accuracy and variance, particularly at the Majumdar-Ghosh point.

Significance. If the comparative claim survives statistical scrutiny, the paper offers a compact and computationally efficient NQS ansatz that is competitive with established baselines and that handles the Majumdar-Ghosh point particularly well. The strengths include public code, benchmarks against ED and DMRG, and honest reporting of failure cases such as the vSineKAN fidelity collapse in the AHM, the poor J2=0.6 results, and the grid-size anomaly in Table II. The main weakness is that the headline L=100 comparison is not yet statistically grounded, as it rests on single-seed, post-hoc tuned runs without error bars.

major comments (4)
  1. [Section VII D and Fig. 11] The L=100 superiority claim is based on a single optimized run per architecture with no seed averaging or error bars. The gaps to the next-best baseline are often only about an order of magnitude, and RNN-LSTM has the best variance at J2=0.3. Without at least a few independent seeds and reported standard deviations, the claim that rSineKAN outperforms RBM, RNN-LSTM, and MLP is not established. Please report seed-averaged means and standard deviations, or an equivalent stability check, for all compared architectures.
  2. [Section VI A and Table IV] The learning rates are chosen per model and per J2 by checking performance against DMRG, and the grid size is changed from 8 to 7 for L=64 because of the anomaly in Table II. This post-hoc selection makes the comparison vulnerable to overfitting to the specific benchmark points. Please describe the selection protocol, including how many hyperparameter configurations were tried and on which J2 values, and ensure that the baselines received an equally thorough tuning; otherwise the head-to-head comparison is not controlled.
  3. [Section VII D and Fig. 9] The DMRG reference energies are stated to use maximum bond dimension chi=200, but no convergence check or truncation error is reported. Since some claimed relative errors are as small as about 1e-5, the DMRG reference may not be converged to that precision for L=100 frustrated chains. Please report the DMRG truncation error or a bond-dimension sweep for at least the L=100 points.
  4. [Section VIII and Section VII C] The Marshall sign rule is encoded into the Hamiltonian for all J2, including J2=0.6 where it is not valid. The paper correctly acknowledges this, but the J2=0.6 results are then not a clean test of the ansatz's ability to represent the true ground state; they are results for a rotated Hamiltonian whose ground state differs from the physical one. Please state this limitation explicitly when interpreting Fig. 12 and the 'best among ansaetze' claim at J2=0.6, or report additional runs without the MSR-based rotation.
minor comments (6)
  1. [Section IX, Fig. 13 caption] The caption says 'linear scaling for RBM and exponential scaling for the RBM,' which appears to be a typo; the text says RNN-LSTM scales linearly and RBM scales exponentially. Please correct the caption.
  2. [Section VII C] The fidelity drop at J2=0.6 is described as 'values around ~0.40%,' which should read approximately 0.40 or 40%; the later text refers to '~41% at J2=0.6.'
  3. [Section VI A] There is a typo in 'All SineKAN architectires are trained'; it should be 'architectures.'
  4. [Section IV] The phrase 'The computational complexity of RBM is more comparable to O(eN)' is ambiguous; please write the intended scaling as O(e^N) or with an explicit statement of the relevant complexity.
  5. [Section II, Eq. (5)] The summation notation 'L,NX' and the use of Almn in the text while the equation uses Amnl are inconsistent; please standardize the index names in Eq. (5) and its surrounding description.
  6. [Section III] The sentence 'fixed learning rate (typically ReLU)' should read 'fixed activation function (typically ReLU).'

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the only self-citation is in architectural motivation and is not load-bearing.

full rationale

We examined the claimed derivation chain. The SineKAN architecture is taken from Ref. [36], whose authors overlap with this paper (Reinhardt and Gleyzer), and the paper cites that work to motivate the choice of sinusoidal activations. This is a minor self-citation, but it is not load-bearing: the paper's central claims are new variational energies, fidelities, and correlations benchmarked against ED and DMRG, computed with VMC; no equation in the paper reduces to a fitted value, and no output quantity is defined in terms of an input. The MSR encoding into the Hamiltonian is a known physics-informed approximation, used consistently for all compared models for J2≤0.5, and the J2=0.6 breakdown is attributed to MSR violation rather than to hidden fitting. Hyperparameters such as learning rates are tuned per model and J2, but they are optimizer settings, not ansatz parameters, and the reported comparisons are empirical benchmarks rather than predictions derived from the tuning. The L=100 superiority claim is a measured result against an external DMRG reference; its dependence on single-seed runs and per-model tuning is a robustness concern, not a circularity concern. Overall, there is no significant circularity in the derivation or benchmarking logic; the score of 2 reflects only the minor self-citation in the architectural motivation.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim depends on empirical hyperparameter choices (learning rates, grid sizes, bias schedules) and on the Marshall sign rule approximation; no new physical entities are postulated.

free parameters (3)
  • Learning rates per model and J2 = 1e-3 to 1e-5 depending on model and J2
    Learning rates were found empirically by scanning values from 1e-2 to 1e-6 against DMRG energies, as described in Section V A and Table IV.
  • Grid size for SineKAN layers = 8 (or 7 for L=64)
    Grid size was set to 8 for most systems but changed to 7 for L=64 to avoid the phase-alignment performance collapse documented in Table II.
  • Zeeman bias h_init = gamma + 0.2, annealed to zero
    For the anisotropic Heisenberg model, a magnetic bias is added and linearly ramped to zero to lift the Neel degeneracy, as described in Section VI.
assumptions (5)
  • standard math Kolmogorov-Arnold superposition theorem
    Used in Section II to justify expressing the wavefunction as a sum of univariate functions.
  • domain assumption SineKAN ansatz expressivity
    Assumes that the specific sinusoidal KAN architecture with finite grid size can approximate the ground-state wavefunctions of the studied spin chains; no rigorous guarantee is provided.
  • domain assumption Marshall sign rule approximation
    For J1-J2 with J2>0, the unitary transformation removing MSR is not exact; the paper relies on the approximation being accurate for J2≤0.5, as stated in Sections IV and VIII.
  • domain assumption Zero total magnetization sector
    For AHM and J1-J2, simulations are restricted to the zero-magnetization sector, which contains the ground state for these Hamiltonians but excludes other sectors.
  • domain assumption DMRG reference accuracy
    DMRG with bond dimension 200 is treated as the exact ground-truth reference for L=32,64,100; for 1D gapped and critical models this is standard but not exact.

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Cite this review

Pith. "Pith review of Probing Quantum Spin Systems with Kolmogorov-Arnold Neural Network Quantum States." pith.science (2026). https://pith.science/paper/L2RIHKXN

@misc{pith2026250601891,
  author       = {Pith},
  title        = {Pith review of: Probing Quantum Spin Systems with Kolmogorov-Arnold Neural Network Quantum States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L2RIHKXN}},
  note         = {Machine review of arXiv:2506.01891}
}
abstract

Neural Quantum States (NQS) are a class of variational wave functions parametrized by neural networks (NNs) to study quantum many-body systems. In this work, we propose \texttt{SineKAN}, a NQS \textit{ansatz} based on Kolmogorov-Arnold Networks (KANs), to represent quantum mechanical wave functions as nested univariate functions. We show that \texttt{SineKAN} wavefunction with learnable sinusoidal activation functions can capture the ground state energies, fidelities and various correlation functions of the one dimensional Transverse-Field Ising model, Anisotropic Heisenberg model, and Antiferromagnetic $J_{1}-J_{2}$ model with different chain lengths. In our study of the $J_1-J_2$ model with $L=100$ sites, we find that the \texttt{SineKAN} model outperforms several previously explored neural quantum state \textit{ans\"atze}, including Restricted Boltzmann Machines (RBMs), Long Short-Term Memory models (LSTMs), and Multi-layer Perceptrons (MLP) \textit{a.k.a.} Feed Forward Neural Networks, when compared to the results obtained from the Density Matrix Renormalization Group (DMRG) algorithm. We find that \texttt{SineKAN} models can be trained to high precisions and accuracies with minimal computational costs.

Figures

Figures reproduced from arXiv: 2506.01891 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Energy gap ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Comparison between the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Top: GS fidelity, Middle: Relative energy error, [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Top: GS fidelity, Middle: Relative energy error, [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Isotropic spin–spin correlation function [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Comparison of the relative error [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Comparison of the variance for the GS energy computed using the [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Comparison of error relative to DMRG [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]

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