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Quantum Variational Activation Functions Empower Kolmogorov-Arnold Networks

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Quantum variational activation functions give Kolmogorov-Arnold networks exponentially wider frequency spectra per parameter.

desk verdict The headline exponential parameter reduction is not established; Theorem 2.2(c) has a load-bearing proof gap, though the empirical study is broad and the framework is clean. read the letter →

arxiv 2509.14026 v2 pith:3SBCFID2 submitted 2025-09-17 quant-ph cs.LG

Jiun-Cheng Jiang , Morris Yu-Chao Huang , Tianlong Chen , Hsi-Sheng Goan This is my paper · ORCID
classification quant-phcs.LG
keywords quantumvariationalactivationfunctiondatare-uploadingcircuitKolmogorov-ArnoldnetworkparameterefficiencyFourierfrequencyspectrumsingle-qubitapproximationNISQ
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

The paper introduces quantum variational activation functions (QVAFs): learnable activation functions implemented by single-qubit data re-uploading circuits, called DARUAN. It claims that adding trainable weights in the data pre-processing stage makes the circuit's accessible frequency support grow exponentially with the number of re-uploading repetitions. With a geometric choice of weights, this yields an exponential reduction in the number of parameters needed to reach a given Fourier approximation error, compared with classical Fourier-series-based KANs. Embedding DARUAN into a KAN gives QKAN, which the authors show empirically can match or beat classical KANs and MLPs on regression, classification, and language modeling while using fewer parameters.

What carries the argument

DARUAN (DatA Re-Uploading ActivaTion): a single-qubit circuit alternating trainable unitaries W^(l) with data encoding gates e^{-i w_l x H}, where H is a Pauli generator and w_l is a trainable pre-processing weight. The spectrum expansion identity Ω_B = {Σ m_l w_l : m_l ∈ {−1,0,1}} is the mechanism that turns linear repetition growth into exponential frequency support; with geometric weights w_l = 2^{l−1}, the maximum frequency becomes K_B = 2^r − 1. This identity, together with the C^m-norm Fourier truncation bound, is what the paper uses to argue exponential parameter efficiency.

What would settle it

Train a DARUAN (geometric weights, r repetitions, single qubit) to approximate a target function whose Fourier coefficients all have equal magnitude, e.g., a square wave or a sum of many harmonics, and record the approximation error as r grows. If the error does not decay like 2^{-r(k+1-m)} or if it saturates far above the truncated-Fourier error for the same maximum frequency, the exponential parameter reduction claim fails. The same test can be run in simulation by comparing against the best possible degree-(2^r - 1) trigonometric polynomial fit.

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Extended reading notes

Core claim

The paper's central claim is that a single-qubit data re-uploading circuit, when each encoding layer is preceded by a trainable weighting w_l, produces a trigonometric polynomial whose frequency set is all sums of the form sum m_l w_l with m_l in {-1,0,1}. Choosing the weights geometrically, w_l = 2^{l-1}, makes the highest reachable frequency 2^r - 1 after r repetitions. Since a classical Fourier KAN must use one parameter per frequency, the paper argues this gives a Θ(log(1/ε)) parameter count to reach approximation error ε in the C^m norm—an exponential reduction over the Θ(ε^{-1/(k+1-m)}) parameters required by Fourier-based KAN. The authors further show empirically that QKANs achieve lo

Load-bearing premise

The paper's exponential parameter-reduction claim assumes that a DARUAN with only O(r) trainable parameters can match the Fourier truncation error of degree 2^r − 1, meaning its restricted coefficient space still contains a trigonometric polynomial as accurate as the full truncated Fourier series.

Editorial extensions

If this is right

  • If the spectrum expansion claim holds, a small number of re-uploading repetitions (r ≈ 5–8) gives QKAN access to the same frequency range as a Fourier KAN with hundreds of parameters.
  • QKANs can be simulated classically because each activation is a single-qubit circuit, so the architecture works on current GPUs and can be trained at GPT-2 scale with roughly one-third the parameters of an MLP baseline.
  • The layer-extension method lets training start with small r and grow the frequency support during training, avoiding the vanishing-gradient problems associated with deep circuits.
  • Trained DARUAN activations can be distilled into classical B-spline or Fourier bases, allowing hybrid quantum-classical deployment where the quantum circuit is only needed for training.

Reading between the lines

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

  • The exponential parameter-reduction theorem as stated presumes that the DARUAN output can actually realize the optimal truncated Fourier series of degree 2^r − 1; with only O(r) trainable parameters the coefficient space is far smaller than the 2^{r+1} coefficients of that series, so the reduction is guaranteed only for the frequency support, not for the approximation error.
  • A direct test is to compare, for a fixed target function, the achieved C^m error of a geometric-weight DARUAN against the theoretical 2^{-r(k+1-m)} bound; the gap will show how much the restricted coefficient set degrades the approximation.
  • The same spectral-identity mechanism should extend to multi-qubit re-uploading circuits, where the frequency set becomes a Minkowski sum of spectra; if so, the exponential-support argument would hold for entangling activations too.
  • Because the single-qubit circuits are efficiently simulable, the practical advantage claimed is parameter efficiency rather than computational speedup; a rigorous complexity separation from classical Fourier KANs would require showing the coefficient restriction is harmless for function approximation, which the paper leaves open.
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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 / 4 minor

Summary. The paper proposes a framework for quantum variational activation functions (QVAFs), instantiated by single-qubit data re-uploading circuits called DARUANs, and embeds these as learnable activations in Kolmogorov-Arnold networks, yielding QKANs. The central theoretical claim is Theorem 2.2: trainable data pre-processing weights w_l expand the accessible frequency spectrum of a DARUAN from O(r) to as many as 3^r frequencies, and with geometric weights w_l=2^{l-1} this yields an exponential parameter reduction relative to classical Fourier-series KANs. The paper also reports extensive experiments on noisy regression, image classification, and autoregressive language modeling, plus a knowledge-distillation scheme from QKANs to classical KANs. The manuscript is well organized and the experimental effort is substantial, but the main theoretical result has a load-bearing gap that invalidates the headline claim.

Significance. If Theorem 2.2 were correct, the paper would demonstrate a striking parameter-efficiency advantage for single-qubit, classically simulable circuits as activation functions. The frequency-support statement (part (b)) is correct and of independent interest, and the empirical results, with released code, suggest that DARUAN/QKAN can be competitive in practice. However, the approximation-error and parameter-efficiency results (parts (c) and (d)) do not follow from the presented proof: the paper shows that the circuit's frequency support is large but never shows that the circuit can realize the coefficients of the optimal truncated Fourier series. Since the circuit has only O(r) trainable parameters, it parametrizes a low-dimensional submanifold of the 2^{r+1}-1 coefficient space, and generic smooth functions require many more parameters. Thus the central theoretical contribution, and the 'exponential reduction' claim in the abstract, are unsupported.

major comments (3)
  1. [§A.2, Theorem 2.2(c)] The proof of part (c) applies the Fourier truncation bound (A.15)-(A.17) to f and immediately concludes ||f−f_B|| ≤ C_f K_B^{-(k+1−m)}. This is valid only if f_B is (or is close to) the optimal truncated Fourier series of f at degree K_B. No such statement is proved or even argued. In fact, f_B is a specific trigonometric polynomial generated by a circuit with O(r) trainable parameters, and a real trigonometric polynomial of degree D=2^r−1 has 2D+1 = 2^{r+1}−1 independent real coefficients. For r≥4, 4r+3 parameters is strictly less than 2^{r+1}−1, so the achievable set is a low-dimensional submanifold of the coefficient space; for generic f∈C^{k+1} the optimal truncation lies outside this model class. The bound in (c) therefore does not follow from Fourier approximation theory.
  2. [Theorem 2.2(d)] Part (d) is directly built on (c): the Θ(log 1/ε) parameter count is obtained by solving C_f 2^{-r(k+1−m)} = ε. Since (c) is unproven, the exponential parameter reduction relative to Fourier KAN is unsupported. The frequency-support statement (b) alone cannot imply expressivity; the number of accessible frequencies is not the same as the number of independently tunable coefficients. This is the headline claim of the abstract and of Section 2.3, so the main theoretical contribution collapses.
  3. [Discussion/Abstract] The abstract and Section 2.3 repeatedly assert 'exponential reduction in parameter size … without loss of expressivity.' Given the gap above, this assertion is not supported by the theory. The manuscript should either provide a proof that the DARUAN class can realize the optimal truncated Fourier coefficients (which seems impossible by dimension counting) or substantially weaken the claim to a statement about frequency support only. As written, the central claim is false or at best unproven.
minor comments (4)
  1. [Theorem 2.2(b) and proof] The statement says '1 ≤ |Ω_B| ≤ (3r − 1)', which appears to be a typo for 3^r − 1. The proof repeats '(3r − 1)'. The correct upper bound from the set of sums Σ m_l w_l with m_l∈{−1,0,1} is 3^r−1 (up to collisions), not 3r−1.
  2. [§2.4, first regression paragraph] The text states 'J0 is a Bessel function that J0(x) = sin(x)/x'. This is incorrect: J0 is the zeroth-order Bessel function of the first kind, not the sinc function. The plot appears to use the true Bessel function, so this is only a typographical error, but it should be corrected.
  3. [Table 4 caption and text] The caption for Figure 3 says HQKAN achieves better perplexity with one-third parameters and 'the same training time', while the text later says '30% less memory during training time'. These should be harmonized. Also, Table 4's partitioning with double midrules is hard to read.
  4. [§A.2 proof of (b)] The proof introduces a '{0} choice' in the sum s_l∈{+1,−1,0}' to account for coincident basis states. This is described in one sentence; a more explicit derivation of how zero appears would aid reproducibility.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the central theorem has an unproven premise but not a circular reduction.

full rationale

The manuscript's derivation chain is largely self-contained and does not reduce to fitted values or to load-bearing self-citations. Theorem 2.1 follows the KAN Fourier-approximation argument with a supplied proof; Theorem 2.2(a) and (b) are direct spectral computations from the single-qubit data re-uploading expansion; the empirical benchmarks are external to the theory. The only serious gap is in Section A.2, proof of Theorem 2.2(c): the text states 'Applying this to f with K=K_A (baseline) or K=K_B (linear layer) proves (c),' but the preceding Fourier-truncation bound (A.15) applies to the optimal truncation g_{≤K}, not to the specific circuit output f_B. Since f_B has only O(r) trainable parameters, it cannot in general equal the degree-(2^r−1) optimal truncation of an arbitrary f, so the exponential parameter reduction in (d) is unsupported. This is an omitted/incorrect step, not a circular equivalence: K_B is not defined in terms of the target error, and no fitted parameter is renamed as a prediction. The self-citations to prior data-re-uploading work are used as background and are not load-bearing. Hence no significant circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the standard KART and Fourier approximation theory, plus one ad hoc assumption: that the DARUAN model class contains the optimal truncated Fourier series. This assumption is load-bearing and is not established by the proof. No new physical entities are introduced; all 'entities' are architectural constructs (DARUAN, QKAN, HQKAN).

free parameters (3)
  • geometric weights w_l = 2^(l-1) = 2^(l-1)
    Chosen by hand in Theorem 2.2(d) to produce integer Fourier frequencies and an exponential K_B; without this specific choice the exponential reduction claim fails.
  • re-uploading depth r = swept over {3,6,...,30} in regressions, fixed at 3 in classification
    Hyperparameter controlling frequency spectrum and parameter count; not predicted, tuned per model.
  • base activation weights w_b, w_d = trained
    Learnable weights for the residual/base activation in Eq. (4.7); these are ordinary trained parameters.
assumptions (4)
  • standard math Kolmogorov-Arnold representation theorem (KART)
    Used as the foundation for KAN architecture (Section 4.1).
  • standard math Fourier approximation theory: smooth functions have C^m truncation error O(K^{-(k+1-m)})
    Invoked in the proof of Theorem 2.2(c) (Appendix A.2).
  • ad hoc to paper The DARUAN circuit class can represent any truncated Fourier series up to degree K_B
    This premise is assumed implicitly in the proof of Theorem 2.2(c) when applying the Fourier truncation bound to f_B. With O(r) parameters, the circuit cannot set arbitrary coefficients, so this assumption is not justified.
  • domain assumption Single-qubit gates can be executed with sufficiently low error on NISQ hardware
    Used to argue experimental feasibility (Section 2.1, citing gate error rates).

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Cite this review

Pith. "Pith review of Quantum Variational Activation Functions Empower Kolmogorov-Arnold Networks." pith.science (2026). https://pith.science/paper/3SBCFID2

@misc{pith2026250914026,
  author       = {Pith},
  title        = {Pith review of: Quantum Variational Activation Functions Empower Kolmogorov-Arnold Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3SBCFID2}},
  note         = {Machine review of arXiv:2509.14026}
}
read the original abstract

Variational quantum circuits (VQCs) are central to quantum machine learning, while recent progress in Kolmogorov-Arnold networks (KANs) highlights the power of learnable activation functions. We unify these directions by introducing the quantum variational activation function (QVAF), a general framework in which parameterized quantum circuits serve as learnable activation functions; in this work we study an efficient single-qubit instantiation called DatA Re-Uploading ActivatioN (DARUAN). We show that DARUAN with trainable data-preprocessing weights can realize an exponentially growing accessible frequency support with the number of re-uploading repetitions; for an explicit geometric choice of these weights, this gives a capacity-level exponential parameter reduction relative to independently parameterized Fourier activations. Embedding DARUAN into KAN yields the quantum-inspired Kolmogorov-Arnold Network (QKAN), which retains the interpretability of the KAN architecture while improving parameter efficiency, expressivity, and generalization. We further introduce layer extension and the hybrid QKAN (HQKAN) architecture to improve scalability and computational efficiency, enabling QKAN modules to act as compact replacements for multi-layer perceptrons (MLPs) in large-scale models. We provide theoretical analysis and extensive experiments on function regression, image classification, and autoregressive generative language modeling, demonstrating the efficiency and scalability of QKANs. Because the single-qubit circuits are efficiently simulable on classical quantum simulators, QKANs have quantum-inspired advantage in parameter efficiency and training stability; DARUANs and QKANs serve as present-day validation of the QVAF concept, and the trained DARUANs are directly executable and feasible on current noisy intermediate-scale quantum (NISQ) hardware for inference validation.

Figures

Figures reproduced from arXiv: 2509.14026 by the authors.

Figure 1
Figure 1. Schematic overview of the proposed quantum machine learning models. (a) A quantum neural network (QNN) serves as a variational activation function (VAF), where an input x is processed by a quantum circuit to yield the output ϕ(x). (b) Specifically, we implement VAFs within a perceptron using a single-qubit data re-uploading circuit with trainable data pre-processing weights wℓ in the ℓ-th encoding block S(wℓx). The … view at source ↗
Figure 2
Figure 2. Function fitting with noise using QKAN and KAN. The target function is f(x) = J0(20x), fitted with noisy data. Both QKAN and KAN use a shape of [1, 1]. The QKAN prediction exhibits smoother behavior compared to KAN which tends to overfit local noise features. We begin with a regression task focused on function fitting. Previous studies have demon￾strated that KANs outperform MLPs in various regression problems, incl… view at source ↗
Figure 3
Figure 3. GPT-2 model trained on the WebText dataset [Radford et al., 2019], incorporating HQKAN and MLP layers. To further reduce parameter count and improve efficiency, we adopt the hybrid QKAN (HQKAN) strategy, where the input and output dimensions fed into QKAN are compressed via fully connected layers, forming an autoencoder-like bottleneck. This compression enables QKAN to operate effectively in a low-dimensional latent… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Knowledge distillation from QKAN to KAN. We consider the regression task f(x, y) = sin(e x + y 2 ). A QKAN is first trained for 500 epochs, after which its learned variational parameters are converted into B-spline coefficients and transferred into a KAN of matching ar…
Figure 5
Figure 5. Figure 5: Empirical noisy function fitting. For each equation, we report the average test RMSE of each model with the best depths of hidden layers. The colored annotations in blue indicate the number of parameters for KANs and in red for QKANs, whence the lower the better. The r…
Figure 6
Figure 6. Figure 6: Visualization of learned QKAN activations for heuristic noisy regression equations. This figure illustrates the per-node activation functions learned by QKANs, similar to the interpretability offered by KANs. We display the QKAN models that achieved the lowest RMSE for…

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.