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Quantum Variational Activation Functions Empower Kolmogorov-Arnold Networks

Jiun-Cheng Jiang, Morris Yu-Chao Huang, Tianlong Chen, Hsi-Sheng Goan

arXiv:2509.14026v2quant-phcs.LG

TL;DR

Learnable activation functions offer expressive, parameter-efficient alternatives, but quantum machine learning has often emphasized data encoding over function approximation. The paper introduces QVAFs and the single-qubit DARUAN architecture within QKANs, showing broad empirical competitiveness with fewer parameters and logarithmic repetition scaling for geometric frequency support.

  • Problem

    Quantum machine learning has focused largely on data encoding, leaving underused the parameter-efficient function-approximation potential of variational quantum circuits.

    Source: §1 · p. 4

  • Method

    The paper introduces QVAFs and implements them through single-qubit DARUAN activations with trainable preprocessing weights embedded in QKANs.

    Source: §1 · p. 4, §2.1 · p. 7

  • Results

    QKANs outperform or rival MLPs and KANs across regression, classification, and generative modeling while using fewer parameters, including about 24% fewer than KANs on average.

    Source: §2.4 · p. 17, §3 · p. 23

  • Takeaways & Limitations

    Trainable preprocessing weights enable geometric frequency support with logarithmically many repetitions, yielding an exponential parameter reduction at the approximation-capacity level.

    Source: §1 · p. 5

    The frequency-accessibility proposition is conditioned on an arbitrary realized preprocessing-weight vector rather than the specific geometric construction.

    Source: Define the model outputs · p. 13

Abstract

from arXiv · show

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.

1 Introduction

The paper introduces QVAFs, using parameterized quantum circuits as learnable activation functions, and instantiates them as efficiently simulable single-qubit DARUAN modules within QKANs. It develops HQKAN for scalability and combines theoretical capacity analysis with empirical validation across regression, classification, and generative modeling.

Framework and instantiation

  • QVAFs generalize learnable activation functions by using parameterized quantum circuits, with DARUAN providing a practical single-qubit instantiation for current hardware and simulation constraints.The broader framework can include multi-qubit entangling circuits, but DARUAN is efficiently classically simulable and executable on current quantum hardware for inference validation.Source: §1 · pp. 4–5

Validation and limitations

  • The framework is evaluated on regression, classification, and generative-modeling tasks, while single-qubit circuits avoid current multi-qubit gate-fidelity, simulation-scaling, and barren-plateau limitations.The paper presents single-qubit DARUAN as a present-day validation platform while leaving entangling multi-qubit QVAF benefits for future hardware and simulation capabilities.Source: §1 · pp. 4–6

QKAN architecture

  • Embedding DARUAN as edge activations yields QKAN, a quantum-inspired KAN framework designed to improve parameter efficiency, expressivity, and generalization while retaining the KAN topology.QKAN modules are simulable on classical accelerators and executable on current quantum hardware for inference validation.Source: §1 · p. 5

Theoretical capacity

  • Trainable data-preprocessing weights give single-qubit re-uploading circuits geometrically spaced frequency support with logarithmically many repetitions, reducing parameters exponentially versus independently parameterized Fourier-series KAN activations.The comparison is made at the approximation-capacity level for matching maximum frequency support.Source: §1 · p. 5

HQKAN architecture

  • HQKAN addresses QKAN’s quadratic parameter growth with input and output dimensions by placing classical compression and expansion layers around a QKAN latent core.This design targets improved scalability and computational efficiency for high-dimensional inputs and large models, including language-model settings.Source: §1 · pp. 5–6

2 Results

This section introduces QVAFs and DARUAN, embeds them into QKAN and HQKAN architectures, and develops theory on their approximation capacity and accessible bandwidth. Experiments across regression, classification, and generative modeling evaluate their performance, efficiency, interpretability, scalability, and knowledge-transfer capabilities.

2.1 The Quantum Variational Activation Function

  • QVAFs replace fixed activations with variational quantum circuits, providing tunable nonlinear transformations that can approximate analytic and, under conditions, arbitrary continuous functions.The framework also accommodates entangling multi-qubit circuits, while DARUAN offers a scalable single-qubit instantiation with spectral compression.Source: §2.1 · pp. 6–8
  • DARUAN instantiates QVAFs as single-qubit data-reuploading activations with trainable preprocessing, alternating encoding and unitaries before Pauli-observable measurement.Its smooth, bounded outputs support stable nonlinear transformations, while single-qubit structure enables efficient classical simulation and direct quantum-hardware inference validation.Source: §2.1 · p. 7

Define the model outputs

  • Layer extension progressively increases DARUAN repetitions, warm-starting higher-frequency refinements from smaller circuits and enabling adaptive stopping based on validation error.This addresses optimization difficulty while jointly controlling spectral capacity, parameter count, and computational cost.Source: §2.1 · p. 7, Define the model outputs · p. 14

2.2 QKAN architecture

  • QKAN replaces each KAN B-spline activation with a single-qubit DARUAN, preserving KAN’s shallow, interpretable structure while providing compact Fourier-like nonlinearities.For depth L, width N, and repetition count r, QKAN parameter count scales as O(N^2Lr), compared with O(N^2LG) for KANs.Source: §2.2 · pp. 8–9

2.3 Theoretical Analysis of QKAN

  • Theorem 2.1 establishes Fourier-series approximation for sufficiently smooth KAN edge functions, with bandwidth K controlling the error rate O(K^−(k+1−m)).Fourier cutoff K plays the role of spline grid resolution G in controlling available modes and approximation accuracy.Source: §2.3 · pp. 9–10

Define the model outputs

  • Theorem 2.2 shows that trainable preprocessing expands accessible frequencies, with geometric weights w_l=2^l−1 yielding filled bandwidth 2^r−1 and Θ(log(1/ε)) repetitions for the benchmark.This is a capacity statement for coefficient-coupled QKAN parameters, not independent Fourier coefficient assignment or a guarantee of optimization convergence.Source: Define the model outputs · pp. 12, 44
  • Proposition 2.3 shows that, for any realized positive preprocessing weights and nondegenerate single-qubit mixing, every nonzero accessible frequency can have a nonzero coefficient.It excludes architecture-imposed cancellation but does not establish coefficient magnitudes, independent assignment, or training convergence.Source: Define the model outputs · p. 13

2.4 Numerical Results

  • QKAN matches or improves KAN and MLP baselines across regression, classification, and generative modeling while providing parameter-efficient, scalable quantum-inspired alternatives.Regression errors fall by up to an order of magnitude in main comparisons; classification remains comparable or better, and HQKAN improves generative-model efficiency.Source: §2.4 · p. 15
  • On the noisy Bessel-function fit, QKAN produces a smoother approximation while KAN captures localized noise features.Both models use shape [1, 1], and the training labels include Gaussian noise with standard deviation 0.1.Source: §2.4 · p. 15, Figure 2 · p. 16
  • QKAN preserves KAN interpretability: symbolic regression recovers the target’s primitives with R2 = 1.000 on three active edges and a pruned-QKAN test MSE of 3.7 × 10−6.The workflow visualizes learned activation functions and transfers QKAN parameters into B-spline representations for matched KAN structures.Source: §2.4 · pp. 15–16
  • QKAN achieves the lowest mean test RMSE on 8 of 10 noisy Feynman functions, using 10% noise and 1,000 training and 1,000 testing points per function.Across the broader benchmark, QKAN achieves the lowest RMSE on over 80% of equations while using 30% fewer parameters than classical KAN on average.Source: §2.4 · pp. 16, 18, Table 1 · p. 16
  • HQKAN achieves competitive image-classification accuracy with substantially fewer parameters, including 70.6% top-5 accuracy on CIFAR-100 using 32,636 parameters, versus 86,948 for MLP and 384,000 for KAN.On CIFAR-10, HQKAN outperforms MLP and KAN in both top-1 and top-5 accuracy with 14,370 parameters; on CIFAR-100, CNN+QKAN achieves the highest top-1 accuracy at 41.2%.Source: §2.4 · p. 19
  • HQKAN-equipped GPT-2 achieves lower final perplexity than MLP while using one-third the parameters and 30% less training memory.Figure 3 compares HQKAN and MLP replacements on WebText, with HQKAN showing superior convergence despite operating through a compressed latent space.Source: §2.4 · p. 19, Figure 3 · p. 20
  • HQKAN reduces GPT-2 memory from 6.5 GB to 4.1 GB while maintaining comparable iteration speed, and HQKANsformer reduces large-scale memory and time by 10% at batch size 800.On one RTX 4090, iteration time is 60 ms versus 63 ms for MLP; on 16×H200, HQKANsformer achieves simultaneous 10% reductions in memory and time.Source: §2.4 · pp. 20, 22, Table 4 · p. 21
  • QKAN-to-KAN knowledge distillation lowers test loss by 70% versus training KAN from scratch under identical conditions.The procedure converts trained QKAN parameters into B-spline bases, followed by limited KAN fine-tuning to compensate for conversion error.Source: §2.4 · p. 22

3 Discussion

QKANs integrate single-qubit DARUAN quantum variational activations into interpretable KANs, delivering parameter-efficient performance across regression, classification, and generative modeling. Layer extension, HQKAN, classical simulation, and hardware validation support scalable deployment while preserving a pathway to quantum inference.

QVAF and DARUAN

  • QKAN integrates the general QVAF concept into KANs through single-qubit DARUAN circuits with trainable preprocessing weights, enabling compact approximation of complex nonlinear mappings.DARUAN’s frequency support grows rapidly with re-uploading repetitions, while shared SU(2)-rotation parameters couple Fourier coefficients and may provide implicit regularization.Source: §3 · pp. 22–23

Overall results

  • Across regression, classification, and generative modeling, QKANs outperform or rival MLPs and traditional KANs with fewer parameters and lower training overhead.These comparisons evaluate QKAN modules within fixed surrounding architectures rather than against full state-of-the-art systems.Source: §3 · pp. 23–24

Scalability extensions

  • Layer extension adaptively increases circuit depth r until a target validation error is reached, reducing unnecessary parameters and computation; HQKAN broadens scalability and task adaptability.The corresponding ablation is reported in Section C.4.Source: §3 · pp. 23–24

Controlled comparisons

  • Under fixed CNN or transformer backbones, QKAN acts as a compact drop-in replacement for MLP or KAN blocks, improving parameter efficiency with comparable or better performance.The controlled comparison isolates the activation module rather than comparing complete architectures with different feature extractors.Source: §3 · p. 24

Ablations and baselines

  • QKAN achieves the lowest or tied-lowest mean RMSE against GeometricFourierKAN and outperforms FourierKAN, while removing DARUAN causes 2–10× RMSE degradation.DARUAN contributes approximately 63–69% of the learned contribution fractions, supporting coefficient coupling—not support size alone—as an empirical advantage.Source: §3 · p. 24

Efficiency and distillation

  • GPU-optimized FlashQKAN has lower mean forward and training-step latency than FourierKAN and substantially lower latency than SplineKAN, while QKAN parameters can initialize KANs for 70% lower test loss.The distillation result is demonstrated on f(x, y) = sin(ex + y2) after converting trained QKAN variational parameters into B-spline coefficients.Source: Figure 4 · p. 23, §3 · p. 24

Hardware validation

  • On ibm_aachen, j0(20x) fitting improved from 3.9439×10^-2 MSE at 1,000 shots to 5.392×10^-4 at 100,000 shots, while 100-shot HQKAN achieved 73.0%/40.0% CIFAR-10/CIFAR-100 accuracy.Exact-solver accuracies were 75.0%/43.0%; differing calibrations and layouts prevent treating the MSE change as controlled shot-scaling evidence.Source: §3 · p. 24

Positioning and limitations

  • Unlike recent multi-qubit QVAF-KAN designs [Werner et al., 2025, Wakaura et al., 2025b,a], single-qubit QKAN avoids their hardware, simulation, barren-plateau, gate-fidelity, and remote-access scalability constraints.Classical simulation remains the preferred practical route, while single-qubit circuits retain direct NISQ inference compatibility; larger entangled DARUAN extensions remain prospective and difficult to validate at scale [Schuld et al., 2021, Pérez-Salinas et al., 2020].Source: §3 · p. 25

4 Methods

The methods develop QKAN by replacing spline-based KAN activations with single-qubit data-reuploading circuits, then add mechanisms for classical transfer, distributed execution, scalable extension, output adaptation, and empirical evaluation.

4.1 Kolmogorov-Arnold Networks

  • KANs approximate multivariate functions through compositions of learnable univariate activations and addition, but B-spline evaluation and grid rescaling can be computational bottlenecks.KANs retain universal approximation and interpretability while typically using smaller widths than comparable MLPs [Liu et al., 2024c].Source: §4.1 · pp. 26–27

4.2 Data Re-Uploading Circuits

  • QKAN replaces each spline activation with a single-qubit data-reuploading circuit that alternates trainable unitaries with scalar encodings w_rx + b_r across r repetitions.This DARUAN construction reduces each edge activation to a scalar-input circuit while retaining trainable preprocessing parameters.Source: §4.2 · p. 27

4.3 Knowledge Distillation from QKANs to KANs

  • Trained DARUAN activations can be sampled and fitted with spline or Fourier coefficients, replacing quantum activations in classical KANs for hybrid deployment.Coefficients may be estimated classically or with HHL, while Fourier replacements support DFT- or QFT-based estimation.Source: §4.3 · pp. 27–28

4.4 Distributed Training of QKANs

  • Because DARUAN activations use independent single-qubit circuits, QKAN layers can run synchronously or asynchronously across distributed quantum machines and support quantum federated learning.A layer with m inputs and n outputs requires m·n independent qubits, while mini-batches can use b·m·n simultaneous qubits or b systems of m·n qubits.Source: §4.4 · p. 28

4.5 QKAN Implementation Methods

  • QKAN implementation uses layer extension to increase repetition depth and frequency capacity, adaptive stopping to control resources, output mappings or scaling to overcome bounded measurements, and SiLU residual paths to stabilize optimization.Layer extension retains the pretrained function by initializing added unitaries to identity, while post-activation strategies broaden output range and preserve QKAN-to-KAN transfer compatibility.Source: §4.5 · pp. 28–29

4.6 Numerical Methods

  • Experiments compare KAN, QKAN, MLP, and HQKAN on Feynman regression, image classification, and GPT-2 language modeling using matched training protocols and task-specific sweeps.HQKAN compresses features to a logarithmic latent dimension around the QKAN block, while VAF-transfer experiments continue KAN training after mapping QKAN activations to B-splines.Source: §4.6 · pp. 31–32

Conflict of Interest · A Supplementary Theoretical Backgrounds · A.1 Proof of Theorem 2.1

The supplementary material discloses editorial conflicts, develops theoretical support for QVAF/DARUAN and QKAN, and proves Theorem 2.1 via trigonometric-polynomial replacement of KAN edge functions. It also distinguishes accessible frequency support from coefficient accessibility and explains how fixed-depth layer composition controls approximation error.

Conflict of Interest

  • The authors disclose editorial roles but state that neither Hsi-Sheng Goan nor Tianlong Chen participated in the journal’s review or decision process.Source: Conflict of Interest · p. 33

A Supplementary Theoretical Backgrounds

  • QVAF is a general activation-function framework, while the supplementary results focus on DARUAN, its single-qubit QVAF instantiation used in QKAN.The framework may also be instantiated by genuine multi-qubit quantum machine-learning circuits.Source: A Supplementary Theoretical Backgrounds · p. 41
  • Theorem 2.2 shows that geometric trainable preprocessing weights yield an exponentially large filled accessible frequency block using only linearly many re-uploading repetitions.This is a capacity statement about accessible support; Fourier coefficients remain coupled through shared single-qubit rotation parameters.Source: A Supplementary Theoretical Backgrounds · p. 41
  • Proposition 2.3 complements the support result by showing that, under a nondegenerate single-qubit mixing assumption, no nonzero accessible frequency is forced to vanish by parameter-independent cancellation.Source: A Supplementary Theoretical Backgrounds · p. 41

A.1 Proof of Theorem 2.1

  • Theorem 2.1 adapts KAN approximation theory by replacing spline approximation with trigonometric-polynomial approximation for each edge function.For each edge function, classical Fourier theory supplies a trigonometric polynomial with bounded frequencies on the relevant compact interval.Source: A Supplementary Theoretical Backgrounds · p. 41, §A.1 · p. 41

A Supplementary Theoretical Backgrounds

A.1 Proof of Theorem 2.1

  • The proof propagates one-layer approximation errors through the fixed KAN representation using local Lipschitz continuity of smooth layer compositions in the C_m norm.Replacing layers successively produces residuals whose finite sum is absorbed into a single constant because the representation depth L is fixed.Source: §A.1 · pp. 41–42

A.2 Proof of Theorem 2.2

This section restates and proves Theorem 2.2, contrasting unweighted circuits with a trainable preprocessing layer that expands accessible frequencies from arithmetic spectra to subset-sum spectra.

A.2 Proof of Theorem 2.2

  • Trainable preprocessing weights ω expand the accessible support from arithmetic spectra of repeated identical encodings to subset-sum spectra controlled by {w_ℓ}^r.This is the proof’s additional step relative to prior spectral analyses of data re-uploading circuits.Source: §A.2 · p. 42
  • The proof establishes Theorem 2.2 on accessible spectra and Fourier approximation capacity for QKANs with trainable preprocessing.The theorem assumes a C^k+1 periodic extension, depth r≥1, and single-qubit generators H = σ_j/2 with frequencies normalized by the eigenvalue gap.Source: §A.2 · p. 42
  • The analysis uses single-qubit generators H = σ_j/2, where σ_j is one of σ_x, σ_y, or σ_z, and measures frequencies in units of H’s eigenvalue gap.Under this normalization, the eigenvalue gap equals one.Source: §A.2 · p. 42
  • The theorem distinguishes a baseline unweighted circuit from a circuit equipped with a preprocessing layer.In the weighted case, ω = (w_1, …, w_r)^⊤ belongs to (0, ∞)^r; each repeated circuit layer has a parameterized unitary W^(ℓ)(θ_ℓ).Source: §A.2 · p. 42

Define the model outputs

The model outputs are characterized by accessible Fourier spectra, approximation capacity bounds, and a geometric weighted construction that yields exponential frequency growth while retaining coupled circuit coefficients.

Fourier approximation benchmark

  • Fourier approximation bounds depend on the filled low-frequency bandwidth K_fill(ω), but they become circuit approximation guarantees only after coefficient realization is established.The bound is a capacity benchmark and does not assume that coupled DARUAN/QKAN parameters independently control every Fourier coefficient.Source: Define the model outputs · p. 47

Capacity comparison with Fourier-series-based KAN

  • Geometric weights w_l = 2^(l−1) fill frequencies through ±(2^r−1), reaching the same bandwidth benchmark with Θ(log(1/ε)) repetitions versus Θ(ε^(−1/(k+1−m))) independent Fourier modes.This is a capacity comparison: QKAN Fourier coefficients remain coupled through shared circuit parameters rather than independently trainable.Source: Define the model outputs · pp. 44, 47

Baseline accessible spectrum

  • The baseline single-qubit circuit accesses integer frequencies from −r through r, with at most 2r distinct nonzero frequencies.Special parameter choices can make some coefficients vanish, so a trained circuit’s actual support may be smaller than this accessible set.Source: Define the model outputs · pp. 43, 45, 46

Preprocessing-layer accessible spectrum

  • Weighted preprocessing produces signed frequencies from ternary sums of the trainable weights, with at most 3^r−1 nonzero accessible frequencies and possible collisions.The actual Fourier support remains contained in the accessible set, so collisions or coefficient cancellations can reduce realized support.Source: Define the model outputs · pp. 43, 46

A.3 Proof of Proposition 2.3 · B Numerical Results Details

Proposition 2.3 proves that every nonzero frequency in the accessible signed-sum set is structurally realizable in the weighted single-qubit circuit, and that generic unitary parameters activate all such coefficients simultaneously. The result concerns coefficient accessibility—not independent coefficient control, coefficient magnitude, or optimization convergence—and differs from average frequency-profile analyses.

A.3 Proof of Proposition 2.3

  • Under arbitrary realized positive weights and nondegenerate trainable mixing blocks with independent Z-phase tags, every nonzero frequency in Ω_B(ω) has a coefficient function that is not identically zero.The proposition conditions on the realized preprocessing weights and assumes each trainable one-qubit block can realize the required nondegenerate mixer.Source: §A.3 · p. 47
  • Generic choices of the unitary parameters make all coefficients of the finitely many nonzero accessible frequencies simultaneously nonzero.Each coefficient is real-analytic and nonzero on a measure-zero exceptional set; finiteness of Ω_B(ω) permits the simultaneous conclusion.Source: §A.3 · pp. 47, 49
  • The proof expands the circuit in a Pauli-transfer path basis, where each path contributes a frequency from its signed sum and a coefficient formed from trainable interleaving-unitary transfer entries.A tagged-mixer subfamily with a fixed Ry(π/3) mixer makes path prefactors nonzero, while independent phase tags give distinct phase characters for distinct paths.Source: §A.3 · pp. 48–49
  • Distinct phase characters are linearly independent, so the aggregated coefficient for any accessible nonzero frequency cannot vanish identically.For each frequency, at least one path produces it; summing the corresponding nonzero prefactors over distinct characters yields a nonzero trigonometric polynomial.Source: §A.3 · p. 49
  • The proposition establishes coefficient accessibility only, not independent coefficient control, a lower bound on coefficient magnitudes, or optimization convergence to a target Fourier series.Thus, a frequency may be structurally accessible while its coefficient remains small for typical parameter draws.Source: §A.3 · pp. 49–50
  • Unlike Barthe and Pérez-Salinas [2024] average frequency-profile results, Proposition 2.3 asks whether any accessible coefficient is structurally forced to vanish for an arbitrary realized weight vector.Repeated identical generators can concentrate average Fourier content near low frequencies, whereas the proposition concerns nonzero analytic dependence on trainable unitary parameters.Source: §A.3 · pp. 49–50
  • The powers-of-two geometric construction uses layer-dependent effective generators as a capacity witness, while empirical sections test whether its structured coefficient coupling benefits optimization and generalization.This construction differs from repeated copies of the same generator and aligns with the exponentially spreading support example discussed by Barthe and Pérez-Salinas [2024].Source: §A.3 · p. 50

B.1 Complementary results of Figure 5 · B.2 Visualization of QKAN activations on noisy function regression

Across 66 symbolic-regression equations, QKAN generally matches or outperforms classical KAN while using fewer parameters, and learned activations remain smooth and structured under input noise.

B.1 Complementary results of Figure 5

  • Table 5 reports RMSE and trainable-parameter counts for all 66 expressions, showing comparable or superior QKAN accuracy with significantly fewer parameters.The table’s bold entries identify the lower-RMSE model in each row.Source: Table 5 · p. 50, §B.1 · p. 50
  • The detailed benchmark includes equations where QKAN has lower RMSE and fewer parameters than KAN, including the expression x1y1 + x2y2 + x3y3.For this expression, the reported RMSE values are 1.476 × 10−03 and 8.556 × 10−04, with parameter counts 595 and 776, respectively.Source: §B.1 · p. 51
  • Other equations show near-matched errors or modest QKAN advantages, supporting the conclusion that parameter reduction does not generally sacrifice approximation accuracy.The reported rows include cases where the two models have identical RMSE and cases where QKAN’s RMSE is lower.Source: §B.1 · p. 51
  • QKAN achieves the best test RMSE on more than 80% of the 66-equation Feynman benchmark, using far fewer parameters than KAN in most cases.Figure 5 compares MLP, KAN, and QKAN at each model’s best hidden-layer depth.Source: Figure 5 · p. 52

B.2 Visualization of QKAN activations on noisy function regression

  • The activation visualizations extend KAN-style interpretability by exposing each learnable one-dimensional function within the QKAN model.The analysis uses learned activation functions from selected noisy regression tasks to inspect internal model structure.Source: §B.2 · p. 53
  • For representative noisy symbolic-regression equations, QKAN learns smooth, structured per-node activation functions despite input noise.Figure 6 visualizes the activations from QKAN models achieving the lowest RMSE on selected benchmark equations.Source: §B.2 · p. 53

B.3 Real-device QKAN/HQKAN inference on IBM Aachen

Real-device IBM Aachen experiments validate the QKAN/HQKAN inference pathway: classification remains close to exact-solver performance at 100 shots, while regression error decreases with shots but retains a hardware-induced gap.

Error sources and scope

  • Within fixed calibration windows, finite-shot uncertainty scales as S−1/2 and its mean-square contribution as S−1, while residual gate, decoherence, readout, and mitigation bias remain at large shot counts.The study therefore treats the experiment as hardware-inference feasibility and robustness validation, not a hardware-speedup claim.Source: §B.3 · p. 53

Function-fit inference

  • Regression test MSE falls from 6.56 × 10−1 at 1 shot to 5.392 × 10−4 at 100,000 shots, versus an exact-solver reference of approximately 2.1 × 10−7.The observed decrease is consistent with improved shot statistics, but differing calibration windows and layouts prevent treating it as a controlled shot-scaling estimate.Source: §B.3 · pp. 55–56, Figure 7 · p. 55

Task-dependent robustness

  • Classification tolerates moderate expectation-value errors because class margins and downstream aggregation can preserve discrete decisions, whereas regression directly penalizes analog pointwise deviations.Top-1 labels remain unchanged when perturbations are smaller than the relevant class margin, while regression MSE continues improving with additional shots.Source: §B.3 · pp. 56–57

Classification inference

  • At 100 shots, HQKAN achieves 73.0%/40.0% top-1 and 98.0%/67.0% top-5 accuracy on CIFAR-10/CIFAR-100, close to exact-solver references of 75.0%/43.0% and 97.0%/68.0%.The hardware results remain usable under present-day NISQ noise, with CIFAR-10 and CIFAR-100 accuracies saturating at 100 shots.Source: Figure 7 · p. 55, §B.3 · pp. 56–57

B.4 Interpretability: the full symbolic-regression workflow

QKAN reproduces the full six-step symbolic-regression workflow for f(x, y) = exp(sin(πx) + y2), recovering the exact ground-truth expression after symbolic fitting and number snapping. Successful recovery depends on refitting the pruned subnet with incremental DARUAN layer extension and frequency-diverse initialization.

B.4 Interpretability: the full symbolic-regression workflow

  • The workflow trains a sparsified QKAN([2, 5, 1]) with L1+entropy regularization, prunes to one hidden neuron, fits primitive functions, refines edge parameters with LBFGS, and composes the result with sympy.The experiment uses 1,000 training and 1,000 test points sampled uniformly from [−1, 1]2.Source: §B.4 · p. 58
  • The pruned QKAN identifies sin on x, x2 on y, and exp on the hidden-to-output edge, each with R2 = 1.000.The learned x-edge frequency snaps to a ≈π, matching the target’s sin(πx) term.Source: §B.4 · p. 58
  • Exact recovery requires training a fresh QKAN([2, 1, 1]) with DARUAN extension r:3→48 and frequency-band-diverse preactivation initialization, rather than naïvely pruning.Naïve pruning leaves MSE > 10−1, whereas the specified procedure recovers sin/x2/exp at machine precision.Source: §B.4 · p. 58
  • The six-step workflow recovers f(x, y) = exp(sin(πx) + y2) exactly, after fitting symbolic edge functions, refining affine parameters, composing them, and snapping numerical constants.Figure 9 reports test MSE 9.6 × 10−5 for QKAN([2, 5, 1]) and 3.7 × 10−6 for the refit QKAN([2, 1, 1]); all three edges achieve R2 = 1.000.Source: §B.4 · p. 58, Figure 9 · p. 59

C Additional Baseline Comparisons and Ablation Studies · C.1 Fourier-based KAN (FourierKAN) baseline comparison · C.2 GeometricFourierKAN comparison

Additional comparisons show that QKAN outperforms FourierKAN across all ten equations, while geometric frequency allocation alone does not explain its advantage. The experiments attribute the remaining benefit to DARUAN’s shared rotation-parameter coupling, which structures Fourier amplitudes and can regularize noisy-label learning.

C Additional Baseline Comparisons and Ablation Studies

  • The regression baselines and ablations use 10% label noise and report mean ± standard deviation across five random seeds.Runtime and precision analyses follow task-specific protocols stated in their respective subsections.Source: C Additional Baseline Comparisons and… · p. 59
  • The section also reports symbolic-form recovery for a pruned QKAN on an exponential-sine regression function.After LBFGS refinement, all three surviving edges match their assigned primitives with R2 = 1.000, and presymbolic test MSE is 3.66 × 10^-6.Source: Table 6 · p. 60

C.1 Fourier-based KAN (FourierKAN) baseline comparison

  • QKAN wins all ten benchmark equations against FourierKAN in test RMSE.Table 7 reports mean ± standard deviation over five seeds; FourierKAN uses grid sizes g ∈ {1, 5, 15}, while QKAN uses reps = 3.Source: Table 7 · p. 60, §C.1 · p. 60
  • Increasing FourierKAN’s grid size does not improve performance and often worsens generalization under noisy labels.The comparison evaluates Fourier series sizes g ∈ {1, 5, 15}.Source: §C.1 · p. 60

C.2 GeometricFourierKAN comparison

  • GeometricFourierKAN tests whether DARUAN’s geometric frequency allocation, rather than quantum parameterization, explains QKAN’s advantage.It uses frequencies {1, 2, 4, . . . , 2^K−1}, whereas FourierKAN uses standard arithmetic frequencies {1, 2, 3, . . . , K}.Source: §C.2 · p. 61
  • QKAN is lowest or tied-lowest in mean RMSE across the reported equations against both GeometricFourierKAN and FourierKAN.Neither classical comparator obtains a lower mean RMSE in the comparison.Source: §C.2 · p. 61, Table 8 · p. 61

C Additional Baseline Comparisons and Ablation Studies

  • The comparisons support shared rotation parameters as the key distinction, because they jointly determine Fourier amplitudes instead of independently assigning coefficients.This structured coupling is presented as an implicit regularizer beyond merely using Fourier-type modes.Source: §C.1 · p. 60, §C.2 · p. 61

C.3 DARUAN contribution ablation (wb vs wd) · C.4 Layer extension ablation

The ablations show that DARUAN is the primary QKAN activation contributor, while progressive layer extension improves optimization for large-repetition models. The classical base activation mainly provides residual stabilization, whereas training the full large model from scratch often yields poorer, more variable solutions.

C.3 DARUAN contribution ablation (wb vs wd)

  • The QKAN edge activation combines a classical base activation and DARUAN through ϕ(x) = wb · σbase(x) + wd · DARUAN(x).The ablation compares four variants across five benchmark equations.Source: §C.3 · p. 61
  • The learned DARUAN contribution fraction is consistently 63–69% across all benchmark equations.The fraction is defined as |wd|/(|wb| + |wd|).Source: §C.3 · p. 61
  • The base activation primarily acts as a residual connection that stabilizes training.This role is consistent with the small performance effect of setting wb = 0 in the benchmark.Source: §C.3 · p. 61
  • Removing DARUAN increases RMSE by 2–10× across the five benchmark equations, whereas removing the base activation has only a small effect.Table 9 reports mean ± std RMSE over five seeds; the learned DARUAN contribution fraction is consistently 63–69%.Source: §C.3 · p. 61, Table 9 · p. 62

C.4 Layer extension ablation

  • Progressive layer extension wins all reported equations in the test-MSE comparison.Table 10 reports mean ± std over five seeds.Source: Table 10 · p. 62
  • The layer-extension schedule uses repetitions {1, 5, 10, 15, 20}, warm-starting from the previous model with 100 steps per stage.It is compared against training the full model from scratch with reps = 20 for 500 total steps.Source: §C.4 · p. 62
  • Training the full large-repetition model from scratch often converges to poorer solutions with higher variance.This supports layer extension as an important optimization strategy for large-repetition QKANs.Source: §C.4 · p. 62

C.5 Runtime analysis · C.6 Parameter and simulation precision sensitivity

FlashQKAN enables fast, architecture-matched QKAN bottleneck execution on GPUs, while QKAN remains comparatively robust to parameter quantization and reduced-precision state-vector simulation. These results support practical, mixed-precision deployment without requiring fp64 simulation.

C.5 Runtime analysis

  • FlashQKAN evaluates single-qubit DARUAN bottlenecks as optimized dense GPU kernels, using fused CuTe kernels with __sincosf intrinsics and warp-shuffle reductions.The benchmark matches the HQKAN-style Linear(100, 10) →KAN-family([10, 10]) →Linear(10, 100) replacement block and uses bfloat16 tensors where supported.Source: §C.5 · pp. 62–63
  • 0.072 ± 0.005 vs. 0.200 ± 0.011 ms forward, and 0.507 ± 0.042 vs. 0.808 ± 0.013 ms per training step, QKAN is faster than SplineKAN in the matched bottleneck benchmark and has lower mean latency than FourierKAN.The comparison uses FlashQKAN’s CuTe backend on an NVIDIA RTX 5090, with batch size 1000 and 100 timed runs after 10 warmups.Source: Table 11 · p. 63, §C.5 · p. 63

C.6 Parameter and simulation precision sensitivity

  • The precision study separately tests classical parameter quantization and quantum-circuit simulation dtype, distinguishing model-weight degradation from state-vector evaluation effects.Models are trained at full precision before 32-bit and 4-bit parameter quantization, while simulation precision is swept using fixed trained weights.Source: §C.6 · pp. 63–64
  • MLP degrades most severely at 4-bit, with an average 3.8× RMSE ratio, while QKAN reaches 2.2× and FourierKAN 1.9×.QKAN’s bounded rotation angles in [0, 2π) limit quantization sensitivity.Source: §C.6 · pp. 63–64, Table 12 · p. 64
  • float64 and float32 agree to six decimals on all three tested equations, showing that fp64 intermediate precision is unnecessary for single-qubit state-vector simulation.The sweep evaluates the same trained QKAN weights through FlashQKAN’s Triton backend across float64, float32, bfloat16, and float8_e4m3fn.Source: §C.6 · p. 64, Table 13 · p. 64
  • Under Triton with pre-scaled state checkpoints, bfloat16 and float8_e4m3fn match the float32 reference within 1.01× on every tested equation.This indicates that reduced-precision QKAN simulation fits standard mixed-precision inference pipelines.Source: §C.6 · p. 64, Table 13 · p. 64

D Experimental Details · D.1 Software implementation. · D.2 Software and system information.

The implementation uses PyTorch-based complex tensor operations and optimized GPU kernels to simulate single-qubit QKAN layers efficiently. Reproducibility is supported by detailed software, dependency, and hardware specifications for the main and supplementary experiments.

D.1 Software implementation.

  • PyTorch extends tensor operations with custom quantum-state evolution routines for CUDA-enabled QKAN simulation.The implementation targets efficient numerical simulation on GPUs.Source: §D.1 · p. 65
  • Quantum states use complex tensors shaped (B, N, M, 2), representing batches, post-nodes, pre-nodes, and single-qubit amplitudes.The final dimension encodes the two amplitudes of the qubit state.Source: §D.1 · p. 65
  • Quantum gates use complex tensors shaped (N, M, 2, 2), indexing node interactions and single-qubit unitary matrices.The first two dimensions index input-output node interactions, while the final dimensions encode the 2 × 2 gate matrix.Source: §D.1 · p. 65
  • The ansatz uses parameterized single-qubit rotations for data encoding and trainable unitaries, with Pauli-Z measurement and Hadamard initialization.Hadamard initialization empirically improves training stability and convergence.Source: §D.1 · p. 65
  • FlashQKAN provides Triton and CUTLASS CuTe GPU backends that fuse or implement the DARUAN recurrence using mixed precision, custom CUDA kernels, and warp-level reductions.Both backends exploit the two-amplitude single-qubit state and map computation to batched GPU operations; the CuTe backend supplies the runtime benchmark implementation in Section C.5.Source: §D.1 · p. 65

D.2 Software and system information.

  • Supplementary FlashQKAN experiments used QKAN 0.2.2, Python 3.13.2, PyTorch 2.11.0+cu130, Triton 3.6.0, CUDA Toolkit 13.0.2, and an NVIDIA RTX 5090.The CuTe JIT build additionally used Ninja 1.13.0, CUDA compilation tools 12.8.93, and CUTLASS/CuTe headers v4.4.0-22-g4ca61d06.Source: §D.2 · p. 65
  • The solver="cute" option and qkan[cute] extra provide build support for the CUTLASS/CuTe-header CUDA backend rather than a separate CuTe Python package.The backend compiles from CUTLASS/CuTe headers during extension build or uses a JIT fallback when those headers are available.Source: §D.2 · pp. 65–66
  • Table 14 summarizes the software versions, dependencies, and hardware specifications used in the main-text experiments.The table is presented as the reproducibility reference for the main experiments.Source: §D.2 · p. 65, Table 14 · p. 67

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