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QuKAN: A Quantum Circuit Born Machine approach to Quantum Kolmogorov Arnold Networks

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that KAN neural-network layers can be implemented as quantum circuits, with B-spline basis functions encoded as quantum amplitudes and read out through Born-rule probabilities.

desk verdict A promising but overclaimed QCBM-based KAN variant: the Born-rule readout restricts the quantum residual to nonnegative spline mixtures, and the paper doesn't address it. read the letter →

arxiv 2506.22340 v1 pith:XQ37JPKN submitted 2025-06-27 quant-ph cs.CVcs.LG

Yannick Werner , Akash Malemath , Mengxi Liu , Vitor Fortes Rey , Nikolaos Palaiodimopoulos , Paul Lukowicz , Maximilian Kiefer-Emmanouilidis This is my paper
classification quant-phcs.CVcs.LG
keywords quantumKolmogorov-ArnoldnetworkcircuitBornmachineB-splinebasisfunctionssuperpositiondistributionlearningvariationalclassifieramplitudeencodinghybridquantum-classicalfunctionapproximation
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

This paper tries to establish that Kolmogorov-Arnold Networks (KANs), whose learnable functions are linear combinations of B-spline basis functions, can be transplanted into quantum circuits without losing their structure or interpretability. The authors encode a family of B-spline basis functions into the amplitudes of a quantum state via a Quantum Circuit Born Machine, split the register into label and position qubits, and read function values by projective measurement. Trainable weights on the label register then play the role of the KAN's learnable coefficients, giving a hybrid quantum-classical residual function and, by absorbing the activation into the superposition, a fully quantum residual. On moons, Iris, and simple regression benchmarks the hybrid model matches a classical rigid-grid KAN and beats variational quantum classifiers, and the pre-training of the spline embedding is shown to be essential. If the construction holds, KANs become a concrete route for interpretable quantum machine learning that exploits superposition to evaluate multiple basis functions at once.

What carries the argument

The label–position register decomposition of a Quantum Circuit Born Machine. The computational basis is split into labelling qubits that index basis functions and position qubits that index inputs; a pre-trained QCBM encodes B-spline basis functions $B_i(x)$ as amplitudes $\beta_i(x)$, and projective measurements at a fixed position $x$ return $p_f(x)=\sum_i |c_i|^2 |\beta_i(x)|^2$. Trainable strongly-entangling layers on the label register supply the coefficients $c_i$, with a classical SiLU branch and scaling weights in the hybrid version. This object carries the whole argument by making the KAN's linear-combination structure literally a quantum superposition.

What would settle it

Take the single-basis residual $f(x) = -B_0(x)$ on a compact interval, train QuKAN with one label to minimize squared error, and check whether the learned probability output matches $-B_0(x)$. Since $p_f(x) = |c|^2 |\beta_0(x)|^2 \ge 0$, the best fit will be $+B_0(x)$-like; a measured error bounded below by the norm of the negative lobe would show the readout cannot represent signed coefficients, contradicting the equivalence claimed in Eq. (9).

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Extended reading notes

Core claim

The central discovery is that the KAN residual function $f(x)=\sum_i \tilde{c}_i B_i(x)$ can be represented by a quantum state $|f\rangle = \sum_i c_i |i\rangle|\beta_i\rangle$, with the B-spline evaluations $\beta_i(x)$ pre-trained into the amplitudes of a QCBM and the coefficients $c_i$ left trainable on the label register. Because the Born rule gives $p_f(x)=\sum_i |c_i|^2 |\beta_i(x)|^2$, the probability readout reproduces the spline combination after absorbing the normalization, and this readout extends to a fully quantum residual that also includes the SiLU activation in the superposition. The paper presents empirical demonstrations on the moons and Iris classification tasks and on linear and logarithmic regression tasks, comparing favourably to variational quantum classifiers and to prior QKAN proposals.

Load-bearing premise

The central assumption is that a KAN residual function, whose coefficients are signed real numbers, can be rebuilt from a probability mixture of squared, normalized spline amplitudes; if the sign information cannot be recovered, the quantum residual is not equivalent to the KAN layer it claims to implement.

Editorial extensions

If this is right

  • KAN layers can be executed on quantum hardware with the same compositional structure, giving interpretable quantum models whose components remain identifiable as basis functions.
  • Multiple basis-function evaluations are obtained in parallel from a single projective measurement, so the function-evaluation cost per layer can be constant in the number of basis functions, limited only by the qubit count.
  • The pre-trained spline embedding transfers across tasks: one QCBM can be trained once on a basis set and reused as the quantum backbone of many different KANs.
  • The fully quantum residual absorbs classical activations into the superposition, removing the hybrid branch entirely and leaving a single quantum circuit as the residual evaluator.
  • Empirical parity with a rigid-grid classical KAN on the tested benchmarks suggests the quantum version's representational bottleneck is the spline basis itself, not the quantum encoding.

Reading between the lines

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

  • The nonnegativity of the Born-rule readout means the quantum residual is effectively a mixture model; the paper's own Eq. (9) identity requires positive coefficients, so a direct sign-carrying generalization would need either a phase or amplitude trick, or an affine shift correction applied outside the circuit.
  • The same label-position decomposition could be used to encode other function bases (wavelets, radial basis functions, or Fourier features) since the QCBM is basis-agnostic; a testable extension is whether Chebyshev or Fourier bases give better quantum KANs than B-splines.
  • The claim that pre-training the spline embedding is decisive could be tested against random but fixed quantum features; the paper's ablation only compares pre-trained splines against an untrained uniform superposition, not against a random fixed feature map.
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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 / 5 minor

Summary. The paper proposes QuKAN, a quantum circuit Born machine (QCBM) implementation of Kolmogorov-Arnold Networks. A QCBM is pre-trained to encode B-spline basis functions into the amplitudes of a register split into label and position qubits; the label register is then trained to weight these basis functions, and the position-register measurement probabilities are used as the function value. The authors present a hybrid architecture (quantum spline residual plus a classical SiLU branch) and a fully quantum architecture (with the SiLU also absorbed into the superposition). Benchmarks are reported on the make_moons and Iris classification datasets and on two regression functions, with comparisons to VQCs, pyKAN, MLPs, a trainable QKAN, and the Enhanced VQKAN from Wakaura et al. An ablation study shows that the QCBM pre-training of the spline basis improves classification accuracy.

Significance. If the representational claim were established, QuKAN would be a novel amplitude-encoding approach to KANs with a plausible route to parallel evaluation of basis functions. The reported toy benchmarks are encouraging: QuKAN matches or exceeds a rigid-grid pyKAN and outperforms the tested VQCs, and the ablation on pre-training is a useful empirical contribution. However, the central equivalence between the quantum residual and the classical KAN residual is not established: the Born-rule readout restricts the function class to nonnegative, normalized mixtures, and the manuscript does not reconcile this with the signed coefficients of classical KANs. The idea is worth pursuing, but the main claim needs to be either corrected to a nonnegative variant or supplemented with a concrete mechanism (e.g., interference via entangling label gates) that restores signed combinations.

major comments (3)
  1. [Quantum representation of the residual functions (Eqs. (8)-(9)) and Full Quantum KAN] The readout in Eq. (8), p_f(x) = sum_i |c_i beta_i(x)|^2, together with the identification in Eq. (9) (tilde c_i = |c_i|^2 and B_i(x) = |beta_i(x)|^2), maps the signed coefficients of the classical KAN residual in Eq. (5) onto nonnegative, normalized probabilities. A superposition of nonnegative squared B-splines cannot represent functions that require cancellation between basis functions, such as f(x) = B_1(x) - B_2(x). The 'Full Quantum KAN' section acknowledges that the Born-rule readout 'has to be normalized and positive' and proposes to correct the normalization and apply a shift, but an additive shift only changes the global offset; it does not restore per-coefficient signs or the shape restrictions of the convex hull of the |beta_i|^2 distributions. Since the abstract and introduction claim that the architecture 'translates' the KAN residual, the representational equivalence is unsupported. The benchmarks may demonstrate a useful nonnegative variant, but not the stated equivalence to KAN.
  2. [Hybrid QuKAN residual function] The forward pass is described as passing the labelling qubits through 'multiple parametrized entangling layers' to optimize the coefficients in the linear combination of basis functions. If these gates are not diagonal, the state after the label unitary is sum_j |j> (sum_i U_ji c_i |beta_i>), and the measured probability is sum_j |sum_i U_ji c_i beta_i(x)|^2, which contains interference cross-terms and is not equal to Eq. (8). If the gates are restricted to diagonal (phase) operations, then only |c_i|^2 is trainable and the model reduces to the positive-cone construction of Eq. (9). The manuscript does not specify which of these two cases is implemented, so the derivation of Eq. (8) is not consistent with the described circuit. This is load-bearing because the expressivity of the quantum residual, and hence the claimed equivalence to KAN, depends on which readout is actually used.
  3. [Results and Methods (network architecture)] The paper defines a single hybrid residual function in the Methods but does not specify the full network topology used for the reported benchmarks: it is not stated how the two input features of the moons dataset are assigned to the position registers of the different residual units, how many residual functions per layer are used, how the residuals are composed into a feed-forward network, or how the final output probability is converted into a classification decision. Without these details, the feasibility and performance claims in Tables 1-3 and Figures 5-6 cannot be reproduced or independently checked. This is a load-bearing omission for the empirical contribution of the paper.
minor comments (5)
  1. [Superposition Distribution Learning, Eq. (2)] The notation psi_k(x) in Eq. (2) is ambiguous: if |psi_k> are computational basis states of the position register, then psi_k(x) should be delta_{k,x}; please clarify the definition of the basis and the meaning of psi_i(x).
  2. [Pre-training has an effect, Fig. 7] The caption refers to 'a random number generator' while the text describes a model with Hadamard gates and a scalable SiLU residual; please align the caption with the text.
  3. [Abstract and Introduction] There are several grammatical issues: 'KANs potential' should be 'KAN's potential'; 'the fully quantum version the entire architecture' should be 'the fully quantum version, the entire architecture'; and 'In this section we are going to introduce' can be tightened.
  4. [Data availability] The manuscript states that 'Data and Code will be made available on reasonable request.' For a simulation-based machine learning paper, making the code publicly available would substantially strengthen reproducibility; please consider releasing it.
  5. [Summary of the results] The abbreviation 'EVKAN' is used in the summary while the rest of the paper uses 'EVQKAN'; please use a single consistent abbreviation.

Circularity Check

1 steps flagged · score 2.0 of 10

One definitional equivalence at Eq. (9): the quantum residual is set equal to the Born-rule readout, so the 'KAN translation' is an encoding choice; the training and benchmarking remain independent.

  1. self definitional [Methods, 'Quantum representation of the residual functions', Eqs. (8)-(9)]
    "p f (x) = Σ_j |⟨ j,x| f ⟩|^2 = Σ_i |c_i β_i(x)|^2 (8) ... In particular, the classical coefficients can be reconstructed as f (x) ∼ p f (x),when ˜c_i = |c_i|^2, and B_i(x) = |β_i(x)|^2 (9)"

    The paper presents the QCBM readout p_f as the quantum analogue of the classical KAN residual f. But f ∼ p_f is not derived: Eq. (9) defines the effective classical spline weights as |c_i|^2 and the effective basis as |β_i|^2, so the reconstructed function is by construction the Born-rule probability output. The signed coefficients c_i of the classical KAN layer in Eq. (5) are replaced by nonnegative, normalized probabilities, and the paper's own 'Full Quantum KAN' paragraph concedes the Born rule readout 'has to be normalized and positive.' Thus the claimed equivalence between the quantum residual and the KAN residual is an encoding assumption, not a reduction from KAN; the numerical results nonetheless remain independent empirical evidence for trainability.

full rationale

The central training and benchmarking chain is not circular: QuKAN is compared against independent baselines (rigid-grid pyKAN, VQCs with different embeddings, MLPs, QKAN, EVQKAN) on moons, Iris, and regression tasks, and the pretraining ablation gives an external falsifiable check. There are no load-bearing self-citations: the cited QCBM, KAN, QKAN, and EVQKAN references are prior work by other groups or standard textbooks, and no 'uniqueness theorem' is imported from the authors' own prior papers. The only definitional shortcut is Eqs. (8)-(9): the quantum readout is equated to the classical residual by setting c̃_i=|c_i|^2 and B_i=|β_i|^2, so any signed-cancellation structure of a genuine KAN B-spline combination is not representable unless compensated post hoc. The paper explicitly flags the positivity/normalization limitation in the 'Full Quantum KAN' section, so the limitation is disclosed rather than hidden. Because this is a representational-equivalence step at the modeling level rather than a fitted prediction, and the empirical claims rest on independent benchmarks, the overall circularity is mild (score 2).

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

The construction rests on standard quantum postulates and on an unstated convexity restriction: Eq. (8) reads out a probability mixture of nonnegative spline basis functions, so signed KAN coefficients from Eq. (5) cannot be represented directly. The main free choices are architecture hyperparameters; no new physical entities are introduced.

free parameters (5)
  • Number of B-spline basis functions per residual = 4
    Chosen by hand for all benchmarks; determines the expressiveness of each quantum residual function.
  • B-spline degree = 2
    Chosen by hand; sets smoothness of the basis functions encoded by the QCBM.
  • Number of residual layers = 1 or 2 depending on the run
    Architecture depth selected per experiment; regression tables compare one and two layers.
  • Position-register discretization resolution = not specified
    Inputs are mapped to 2^n_x equally spaced points according to Eq. (10); the actual number of position qubits is not reported, so the quantization error is uncontrolled.
  • Residual scaling weights w_f and w_s = trainable
    Output scaling weights in Eq. (12); they are standard trainable model parameters, listed for completeness.
assumptions (5)
  • standard math Kolmogorov-Arnold representation theorem and its generalization to arbitrary-width KANs justify representing multivariate functions via sums of univariate splines.
    Invoked in the Methods summary of classical KAN; the paper builds QuKAN on this foundation.
  • domain assumption B-spline basis functions are nonnegative and can be learned as probability distributions by a QCBM.
    The QCBM pretraining targets normalized B-spline values as probabilities; this requires the basis functions to be nonnegative and normalizable.
  • domain assumption The Born rule allows interpreting squared amplitudes as probabilities and measuring position-register values to evaluate functions.
    Used throughout Eqs. (2) and (8) for the readout of basis function evaluations.
  • domain assumption A QCBM with strongly entangling layers trained by MMD loss can approximate the target B-spline distributions accurately enough.
    The whole method depends on the pretrained QCBM faithfully encoding the basis; no error analysis or convergence guarantee is given.
  • ad hoc to paper Function values can be encoded as probabilities and rescaled without loss, ignoring sign and convexity restrictions.
    Eq. (9) sets f(x) ~ p_f(x) and c_i tilde = |c_i|^2, which forces nonnegative coefficients; this restriction is not stated as a limitation.

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Cite this review

Pith. "Pith review of QuKAN: A Quantum Circuit Born Machine approach to Quantum Kolmogorov Arnold Networks." pith.science (2026). https://pith.science/paper/XQ37JPKN

@misc{pith2026250622340,
  author       = {Pith},
  title        = {Pith review of: QuKAN: A Quantum Circuit Born Machine approach to Quantum Kolmogorov Arnold Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XQ37JPKN}},
  note         = {Machine review of arXiv:2506.22340}
}
read the original abstract

Kolmogorov Arnold Networks (KANs), built upon the Kolmogorov Arnold representation theorem (KAR), have demonstrated promising capabilities in expressing complex functions with fewer neurons. This is achieved by implementing learnable parameters on the edges instead of on the nodes, unlike traditional networks such as Multi-Layer Perceptrons (MLPs). However, KANs potential in quantum machine learning has not yet been well explored. In this work, we present an implementation of these KAN architectures in both hybrid and fully quantum forms using a Quantum Circuit Born Machine (QCBM). We adapt the KAN transfer using pre-trained residual functions, thereby exploiting the representational power of parametrized quantum circuits. In the hybrid model we combine classical KAN components with quantum subroutines, while the fully quantum version the entire architecture of the residual function is translated to a quantum model. We demonstrate the feasibility, interpretability and performance of the proposed Quantum KAN (QuKAN) architecture.

Figures

Figures reproduced from arXiv: 2506.22340 by the authors.

Figure 1
Figure 1. Sketch of the Quantum Circuit Born Machine learning algorithm: Starting from the state |0⟩ Nn we process through a quantum circuit containing strongly entangling layers, so parametrized rotations as well as CNOT gates. The full computational basis is measured via projective measurements and the squared maximum mean discrepancy (MMD) loss is calculated. Note that the comparison here goes along the normalized target d… view at source ↗
Figure 2
Figure 2. Sketch of the training of the QCBM for parallel superposition learning of two target functions. The process follows the same algorithm as shown in figure [1]. The difference lies in the comparison for the optimization. Here we compare to the state representing both functions as given in the Kronecker basis denoted by π˜(x). Hybrid KAN Residual Functions In this section we are going to introduce a hybrid formulation … view at source ↗
Figure 3
Figure 3. QCBM probability output for normalized B-spline basis functions on discretized input interval Hybrid QuKAN residual function To understand how information propagates through the QuKAN architecture, we begin by analysing the processing of a single hybrid residual function. This unit combines a Quantum Function Evaluator (QFE), trained to approximate a set of pre-trained basis functions, with a classical non-linear tr… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Architecture of a single Hybrid QuKAN residual function. The input data gets processed through a classical (the SiLU function) and a quantum transfer. The quantum transfer is based in the pre-trained QCBM Spline function encoding and optimizes the prefactors of the lin…
Figure 5
Figure 5. Figure 5: Decision boundaries on the moons dataset (with noise=0.1) for different models competing quantum models, as well as the classical pyKAN. For pyKAN we introduced a rigid grid and limited the maximum number of splines to 4 in order to ensure a fair comparison with the Qu…
Figure 6
Figure 6. Figure 6: Comparison of the true against the predicted values for the regression task of the QuKAN. The red line indicates perfect prediction. Model Sum Abs. Dist. Avg. Sum Abs. Dist. Med. Sum Abs. Dist. Min. Sum Abs. Dist. Max. QuKAN (2 layers) 0.7524 0.5451 0.0091 3.3094 QuKAN…
Figure 7
Figure 7. Figure 7: Comparison of the QuKAN training accuracy with a VQC architecture and a random number generator. Summary of the results The presented benchmarks demonstrate that the hybrid Quantum Kolmogorov Arnold Network (QuKAN) is a viable implemen￾tation of the classical KAN. On b…
Figure 8
Figure 8. Figure 8: Decision boundaries of the Variational Quantum Classifier with different setups: a) Amplitude Embedding, b) Amplitude Embedding including a total of 4 ancillas. The dataset is the moons dataset with a noise of 0.1. Model Noise level 0.2 0.3 0.5 QuKAN 93.48% 89.44% 83.6…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quantum Variational Activation Functions Empower Kolmogorov-Arnold Networks

    quant-ph 2025-09 reject novelty 5.0 of 10

    QKANs show strong empirical performance on regression, vision, and language tasks, but the claimed exponential parameter reduction is not rigorously established.

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