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REVIEW 3 major objections 8 minor 33 references

Optimization by VarQITE on Adaptive Variational Quantum Kolmogorov-Arnold Network

T0 review · 3 major / 8 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proposes training an Adaptive Variational Quantum Kolmogorov-Arnold Network through Variational Quantum Imaginary Time Evolution (VarQITE), and reports that the pure damping of quantum states—no separate gradient-based…

desk verdict New combination of VarQITE and Adaptive VQKAN, but the target-independent evolution dynamics leave the central claim unsupported. read the letter →

arxiv 2506.22801 v1 pith:BSSWKLWZ submitted 2025-06-28 quant-ph

Hikaru Wakaura , Rahmat Mulyawan , Andriyan B. Suksmono This is my paper · ORCID
classification quant-ph MSC 81P6868Q12 PACS 03.67.Ac03.67.Lx
keywords VarQITEQuantumimaginarytimeevolutionAdaptivevariationalKolmogorov-ArnoldnetworkmachinelearningFunctionfittingclassificationalgorithm
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

Quantum imaginary time evolution is normally a tool for finding ground states; this paper tries to establish that it can instead serve as the training algorithm for a quantum machine learning model. The central claim is that Variational QITE (VarQITE) optimizes an Adaptive Variational Quantum Kolmogorov-Arnold Network: damping the ansatz state by imaginary-time steps drives the network's spline parameters so that the readout expectation tracks the target function, without any external gradient-based optimizer. On toy regression targets and a 2-D classification problem, the authors report that prediction error decreases with imaginary time and that the method beats a quantum neural network baseline on several of the fitted functions. If true, this matters because it would make a physical process—state damping—a general training mechanism for quantum machine learning, and it would extend the QITE family from eigenvalue problems to prediction on multiple input samples.

What carries the argument

The load-bearing object is the VarQITE loss (Eq. 5, summed in Eq. 8), which measures, for each sample, how faithfully the current variational state reproduces one step of imaginary-time damping: $$l_{\mathrm{VarQITE}}^m = \left|1 - \frac{\langle \Psi_\tau | $e^{{-\delta\tau H}}$ | \Psi_{\tau-1} \rangle}{\sqrt{\$\cosh$^2\delta\tau - \$\sinh$^2\delta\tau\,\langle\Psi_{\tau-1}|H|\Psi_{\tau-1}\rangle}}\right|,$$ with the propagator Taylor-expanded (Eq. 6) and reduced to the $\cosh\delta\tau - H\sinh\delta\tau$ identity (Eq. 7) when $H$ is a single Pauli operator. The ansatz being trained is the Adaptive VQKAN of Eqs. (1)–(3), whose per-layer operator strengths $\varphi^{np}_{jk}(n x_m)$ are built from B-spline functions and a Fermi–Dirac-like activation $E_f$; the supervised target $L$ (Eq. 4) is the absolute distance between the Hamiltonian expectation $\langle H\rangle$ and the aimed value. The purpose of the machinery is to replace an external optimizer with a physical contraction: imaginary-time damping drives the state toward lower-energy configurations, and the paper claims this contraction is what moves the readout toward the target function.

What would settle it

Run the same Adaptive VQKAN, ansatz-growth rule, initialization, and COBYLA budget on the five fitting targets, but minimize the supervised loss $L$ of Eq. (4) directly instead of the VarQITE loss of Eq. (8), and compare test-point sums of absolute distances. If the direct-$L$ version matches or beats VarQITE, the claim that imaginary-time damping is the operative training mechanism is falsified; alternatively, exhibiting a single sample where $L_{\mathrm{VarQITE}}$ decreases while $L$ increases would directly contradict the assumed link.

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Extended reading notes

Core claim

The paper's claim, stated in its concluding section, is that VarQITE can optimize Adaptive VQKAN, and it presents this as the first practical application of QITE-family methods to quantum machine learning over multiple input samples. In the method, each training step propagates the variational state under the damping operator $e^{-\delta\tau H}$ with $\delta\tau = 0.1$, growing the adaptive ansatz every $\tau = 2.0$ and increasing the spline grid every $\tau = 0.1$, and the parameters are tuned so that the state best emulates one imaginary-time step; the readout Hamiltonian expectation is then compared with the target value. The reported results are that the supervised loss and the sum of absolute test distances decrease as $\tau$ grows, that the minimum distances on the target function of Eq. (9), the logarithmic function, and the fractional function are smaller than the average of a quantum neural network baseline, and that the fractional-function minimum is also smaller than the Enhanced VQKAN comparison quoted from the prior paper. The paper concedes that convergence is slow, that the distances rebound for $\tau > 6$, and that classification errors remain larger than those of ordinary Adaptive VQKAN, attributing the rebound to the difficulty of emulating imaginary-time propagators accurately.

Load-bearing premise

Everything rests on the assumption that pushing the VarQITE fidelity loss (Eq. 8) down also pushes the actual prediction loss $L$ (Eq. 4) down; the paper never proves or analyzes the relationship between these two different objectives, so the fitting claim is supported only by the empirical correlation seen in a few toy runs.

Editorial extensions

If this is right

  • VarQITE becomes a viable training mechanism for the Adaptive VQKAN family, meaning quantum machine learning can proceed by imaginary-time damping without special training techniques.
  • The same training rule transfers to other VQKAN variants and quantum neural networks, since the damping update does not depend on the specific ansatz form.
  • On the target function of Eq. (9), the logarithm, and the fractional function, the minimum test distances are smaller than the average quantum neural network baseline; the fractional function also beats the Enhanced VQKAN comparison quoted from the prior paper.
  • Slow convergence and the rebound of absolute distances for $\tau > 6$ are the identified open problems, pointing to the fidelity of emulating imaginary-time propagators as the limiting factor.
  • In the 2-D classification setting the damping still reduces the losses, but the errors stay above ordinary Adaptive VQKAN, so more imaginary time is expected to be needed for practical accuracy.

Reading between the lines

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

  • The paper leaves the link between $L_{\mathrm{VarQITE}}$ and the supervised loss $L$ unexamined; a testable extension is to compute both gradients on a single sample and check whether the VarQITE descent direction ever opposes the $L$ descent direction.
  • If the claimed mechanism holds, the natural conjecture it implies is that any quantum machine learning ansatz whose loss is a Hamiltonian expectation could be trained by imaginary-time damping, and that conjecture is testable by swapping the Adaptive VQKAN ansatz for a plain hardware-efficient ansatz on the same five targets.
  • The observed rebound at $\tau > 6$ is attributed to imperfect emulation of the propagator; flipping the sign of $H$ or replacing VarQITE with probabilistic imaginary time evolution on the same schedule would isolate whether the rebound is a property of the damping dynamics or of the variational emulation.
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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

3 major / 8 minor

Summary. The paper proposes training an Adaptive Variational Quantum Kolmogorov-Arnold Network (VQKAN) by Variational Quantum Imaginary Time Evolution (VarQITE). The method updates ansatz parameters by minimizing a VarQITE fidelity loss under a fixed Hamiltonian, with the readout given by the expectation value of the same Hamiltonian. Simulations are reported for regression of five elementary functions and for a two-dimensional classification task. The authors observe decreasing supervised losses and conclude that VarQITE can optimize Adaptive VQKAN, while acknowledging slow convergence, rebounds, poor classification results, and a possible sign problem. The claimed accuracy advantage over a quantum neural network is based on a previous paper rather than on comparisons in this manuscript.

Significance. The idea of using imaginary-time damping as a training signal for quantum machine learning is interesting and, if it worked as described, would offer a non-gradient optimization alternative for small-parameter ansatze. The paper is candid about its limitations and provides enough protocol detail to identify the key equations. However, the manuscript does not establish that minimizing the VarQITE loss reduces the supervised loss, and Eq. (5) is algebraically incorrect for the multi-term Hamiltonian actually used. The empirical results are suggestive but lack error bars for most curves, direct baseline comparisons, and code. As written, the central claim is therefore not yet supported.

major comments (3)
  1. [Section 2, Eqs. (4)-(8)] The central claim that VarQITE optimizes Adaptive VQKAN is not supported because no relationship is established between L_VarQITE and the supervised loss L. With the Hamiltonian H=Z0Z1+Z2Z3 used in Sec. 3.1, exact imaginary-time evolution obeys d<H>/dτ = -2 Var_Ψ(H) ≤ 0, so every input state's readout expectation moves monotonically toward the ground-state expectation of the same fixed H. Since the targets in Eq. (9) range up to e^2≈7.39 while <H>∈[-2,2], the supervised loss |<H>-f_aim| must increase for target values above the current expectation. The sample weights a_m depend on f_aim, but they only rescale the per-sample VarQITE losses; they do not make the evolution target-directed. The observed decrease of L in Sec. 3 therefore requires an additional mechanism (e.g., trainable θ_j in H, a target-dependent H, or optimizing L directly instead of L_VarQITE) that is not described. Please provide an analysis of when minimizing L_VarQITE reduces L, or modify the algorithm so that the evolution is guided by the targets.
  2. [Eq. (5) and (7)] The normalization in Eq. (5) is incorrect for the Hamiltonian used in this paper. The identity e^{-δτH}=cosh δτ - H sinh δτ and the denominator sqrt(cosh^2 δτ - sinh^2 δτ <H>) apply only when H^2=I. For H=Z0Z1+Z2Z3, H^2=2I+2Z0Z1Z2Z3, so Eq. (5) does not compute the fidelity between Ψ_τ and the imaginary-time-propagated state. Consequently the quantity actually minimized in the simulations is not the stated VarQITE loss. The authors should either restrict H to a single Pauli operator or derive the correct normalization for the multi-term Hamiltonian.
  3. [Section 3.1 and Abstract] The abstract's claim that 'our method was more accurate than a quantum neural network in solving some problems' is not supported by any comparison in this manuscript. The only evidence is the statement that, according to the previous paper [28], the minimum test losses for Eq. (9), the logarithmic function, and the fractional function are smaller than the average of a Quantum Neural Network. No baseline results, experimental settings, or statistical tests are shown here, and the classification experiments are explicitly reported to be worse than ordinary Adaptive VQKAN. Please include direct comparisons in the present paper or qualify the accuracy claim to what is actually demonstrated.
minor comments (8)
  1. [Section 1] The sentence 'VQKAN is the quantum version of VQKAN' should read 'quantum version of KAN'.
  2. [Section 2] The sentence 'We use blueqat SDK30 for numerical simulation of quantum calculations and COBYLA of scipy to optimize parameters but to declare the use of others' is incomplete and should be rephrased.
  3. [Eq. (8)] The phrase 'the same as that of eq. 5' is incorrect because Eq. (5) does not define a_m; the cross-reference should be to Eq. (4) or to the surrounding text.
  4. [Section 2] The notation 'nxm' and '1xm' is confusing; the dimension of the input vector and the meaning of the superscript m should be defined once in the text.
  5. [Figures 1-3] Most curves have no error bars and it is not stated whether they are single runs or averages; please specify this and label the y-axes clearly.
  6. [Section 3] The 'sum of absolute distances' on test points is used as a metric but never formally defined; please define this test loss in Section 2 or 3.
  7. [Eq. (2)] The argument of 'acos' may leave the interval [-1,1] for arbitrary parameter values; please explain how this is handled in the implementation.
  8. [Section 1] The paper cites [25] as the Adaptive VQKAN paper but does not clearly state what is new in the present work beyond replacing the optimizer; please add an explicit statement of the incremental contribution.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity in the core VarQITE-VQKAN derivation; the weak links are missing proof and technical normalization errors, not by-construction equivalence.

full rationale

The derivation chain is not circular. In Section 2, Eq. (4) defines the supervised loss L as the absolute distance between the readout expectation <H> and the target f_aim, while Eqs. (5)-(8) define a separate VarQITE objective that tracks imaginary-time propagation under H. Parameters are optimized with COBYLA against L_VarQITE; the predicted values are not obtained by fitting or renaming the training targets. The target enters only through the sample weights a_m in Eq. (8), and L_VarQITE is not algebraically identical to L, so the reported test loss is not forced by construction. The fixed-H concern is a real missing-mechanism/correctness gap rather than circularity: the paper does not define H in terms of f_aim or relabel a fitted parameter as a prediction. Eq. (5)'s denominator assumes H^2=I and is invalid for H=Z0Z1+Z2Z3; this is a technical flaw, not a circular reduction. Self-citations [25] and [28] supply the adaptive ansatz and baseline numbers, but they are not used as an unverified uniqueness theorem, and the baseline comparisons are externally checkable. The manuscript itself concedes limitations (weak classification, possible sign problem, rebound for tau>6), which weaken the empirical claim but do not show that the prediction is equivalent to the input by construction.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The method introduces several hand-chosen hyperparameters and relies on an unproven relationship between the VarQITE objective and the supervised loss. No independent empirical verification is provided beyond the paper's own simulations, so the central claim is supported primarily by the authors' own demonstrations.

free parameters (7)
  • imaginary time step delta_tau = 0.1
    Chosen in Section 2; all results depend on this discretization.
  • initial ansatz hyperparameters = N=10, Nl=1, Nq=4, Ng=8, Ns=8
    Set in Section 2; no ablation or sensitivity analysis.
  • ansatz growth and grid schedule = grow ansatz per 2.0 tau, increase Ns by 4 per 0.1 tau
    Hand-selected in Section 2; directly controls expressiveness and convergence.
  • initial random perturbation = uniform 0 to 0.1
    Added before each VarQITE step; unspecified seed, affects reproducibility.
  • Hamiltonian for fitting = Z0Z1 + Z2Z3
    Chosen in Section 3.1; used both as observable and imaginary-time generator without justification.
  • convergence threshold = 1e-16
    Termination criterion in Section 2; may cause overfitting or non-convergence.
  • sample weighting coefficient a_m = (1 - f_aim + 1/N)/(2 + 1/N)
    Defined in Section 2; couples target values to the VarQITE loss, a hand-selected formula.
assumptions (5)
  • standard math The Taylor expansion of the imaginary-time propagator converges, and for a single Pauli H, e^{-delta_tau H} = cosh(delta_tau) - H sinh(delta_tau).
    Used in Eqs. (6)-(7) to make the propagator computable.
  • domain assumption The parameterized quantum state can represent the target functions with sufficient accuracy.
    No expressibility analysis is given; success depends on the ansatz capacity.
  • ad hoc to paper Minimizing the VarQITE loss L_VarQITE reduces the supervised loss L.
    Eqs. (5) and (8) define a different objective from Eq. (4); the paper does not prove the relationship.
  • domain assumption Swap test and Hadamard test can estimate the overlap in Eq. (5) without error, and COBYLA converges to the optimum.
    Infinite shots are assumed; classical optimizer behavior is taken for granted.
  • ad hoc to paper The adaptive ansatz growth rule preserves the validity of the VarQITE update.
    The ansatz changes during optimization; the effect on imaginary-time fidelity is not analyzed.

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Cite this review

Pith. "Pith review of Optimization by VarQITE on Adaptive Variational Quantum Kolmogorov-Arnold Network." pith.science (2026). https://pith.science/paper/BSSWKLWZ

@misc{pith2026250622801,
  author       = {Pith},
  title        = {Pith review of: Optimization by VarQITE on Adaptive Variational Quantum Kolmogorov-Arnold Network},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BSSWKLWZ}},
  note         = {Machine review of arXiv:2506.22801}
}
read the original abstract

Quantum imaginary time evolution (QITE) is a powerful method to derive the ground states of the systems. Only the damping of quantum states leads it; hence, reaching the ground state is guaranteed by nature without any external manipulation. Numerous QITE methods by many groups are used to improve speed and accuracy, derive excited states, and solve combined optimization problems. However, the QITE methods have not been used for quantum machine learning to predict the ideal values for multiple input values. Therefore, we propose a method for applying QITE methods for quantum machine learning and demonstrate fitting problems of elementary functions and classification problems on a 2-D plane. As a result, we confirmed that our method was more accurate than a quantum neural network in solving some problems. Our method can be used for other quantum machine learning algorithms; hence, it may be the milestone for applying QITE to quantum machine learning.

Figures

Figures reproduced from arXiv: 2506.22801 by the authors.

Figure 1
Figure 1. Imaginary time τ vs. sum of absolute distances (loss function of test points) on the fitting of eq. (9), the radius of the sphere, logarithmic function, fractional function, and exponential function [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Imaginary time τ vs. loss function and loss function without coefficie nts on the fitting of eq. (9), the radius of the sphere, logarithmic function, fractional function, and exponential function [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Imaginary time τ vs. L VarQIT E on fitting of eq. (9), the radius of the sphere, logarithmic function, fractional function, and exponential function. 4/9 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: (Left) Imaginary time τ vs. the average and median for 5 attempts of the sum of absolute distances on the fitting of eq. (9). The error bar indicates the standard deviation. (Right) Imaginary time τ vs. loss function and loss function without coefficients for 5 attempt…
Figure 5
Figure 5. Figure 5: Imaginary time τ vs. sum of absolute distances (loss function of test points) on a classification problem [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Imaginary time τ vs. loss function and loss function without coefficients on a classification problem [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: Imaginary time τ vs. L VarQIT E on classification problem. 6/9 [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: (Left) Imaginary time τ vs. the average and median for 5 attempts of the sum of absolute distances on a classification problem. (Right) Imaginary time τ vs. loss function and loss function without coefficients for 5 attempts of the sum of absolute distances on a classi…

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