Generative Adversarial Variational Quantum KAN

• The framework integrates variational quantum Kolmogorov-Arnold networks with adversarial training to deliver parameter-efficient image synthesis.
• It employs trainable quantum circuits with spline-parameterized Ry rotations to achieve universal function decomposition and enhanced expressivity.
• Experiments on MNIST and CIFAR-10 demonstrate competitive accuracy and sample diversity under low-data regimes with far fewer parameters than classical CNNs.

A Generative Adversarial Variational Quantum Kolmogorov–Arnold Network (GAVQKAN) is a quantum-enhanced adversarial generative modeling framework in which the generator is implemented as a variational quantum Kolmogorov–Arnold network (VQ-KAN) and the discriminator is typically classical, often a lightweight convolutional neural network. The approach leverages the Kolmogorov–Arnold representation for universal function decomposition and embeds it within a variational quantum circuit ansatz to achieve competitive sample quality with high parameter efficiency—outperforming neural network and standard quantum GAN baselines in limited-data regimes for tasks such as image synthesis on MNIST and CIFAR-10 (Wakaura, 11 Dec 2025).

1. Theoretical Foundation: Kolmogorov–Arnold Decomposition in Quantum Networks

The classical Kolmogorov–Arnold Network (KAN) is based on the theorem that any continuous multivariate function $f: \mathbb{R}^n \to \mathbb{R}$ can be expressed as

$$f(x) = \sum_{k=1}^{2n} \phi_k \left( \sum_{j=1}^n \psi_{jk}(x_j) \right),$$

where $\psi_{jk}$ and $\phi_k$ are univariate nonlinear maps. In the VQ-KAN setting, both the inner ($\psi_{jk}$) and outer ($\phi_k$) maps are parametrized and realized implicitly by trainable quantum circuits ("quantum neurons"). Inputs $z \in \mathbb{R}^n$ are encoded into a quantum state via $N_q$ single-qubit Ry rotations: $$|\Psi_{\text{ini}}(z)\rangle = \bigotimes_{j=0}^{N_q-1} Ry_j(z_j)\,|0\rangle^{\otimes N_q}$$ with $z_j$ normalized to $[0, 2\pi]$. Each VQ-KAN layer comprises $N_d$ sub-layers, each applying a pattern of parametric Ry gates (with angles governed by spline-parameterizations, e.g., $c_s^{n,j,d}$ at quantized grid points) and nearest-neighbor entangling CZ gates. The measurement yields a $2^{N_q}$-element probability vector, which is downsampled or averaged for use in subsequent layers or as final output.

2. GAVQKAN GAN Integration: Architecture and Training

In GAVQKAN, the generator is one or more VQ-KAN blocks, each operating on a random latent vector $z \in \mathbb{R}^{N_q}$ to produce a sequence of measured Born probabilities. For 32×32 (CIFAR-10) or 16×16 (MNIST) images, several VQ-KAN blocks are used in parallel to generate patches, which are stitched to reconstruct the overall image. Each block typically uses $N_l=1$ layer, $N_d=8$ sublayers (circuit depth), $N_q=8$ qubits, 4 spline segments per input variable, and 8 input grid points per spline, leading to 256 trainable parameters per block. The measured probability vector is interpreted as a grayscale image patch. The discriminator is a classical CNN (typically 3 fully connected layers, e.g., [256, 32, 1] for a 16×16 patch) with ReLU activations and sigmoid normalization.

Training proceeds by alternating Stochastic Gradient Descent (SGD) steps on the generator and discriminator. Quantum gradients (for VQ-KAN circuit parameters) are computed with the parameter-shift rule: $$\frac{\partial \mathcal{L}}{\partial c_s^{n,j,d}} = \frac{1}{2}\left[\mathcal{L}(c_s^{n,j,d} + \frac{\pi}{2}) - \mathcal{L}(c_s^{n,j,d} - \frac{\pi}{2})\right]$$ as implemented in PennyLane (Wakaura, 11 Dec 2025).

3. Loss Functions and Optimization

The adversarial loss follows the standard Goodfellow GAN formulation: $$\min_G \max_D V(D,G) = \mathbb{E}_{x \sim p_\text{data}}[\log D(x)] + \mathbb{E}_{z \sim p_z}[\log(1-D(G(z)))]$$ specifically,

• Discriminator: $L_D = -[\log D(x) + \log(1 - D(G(z)))]$
• Generator: $L_G = -\log D(G(z))$

No explicit quantum-specific regularization is required; the adversarial signal suffices to train the VQ-KAN generator. Optimization in the benchmark is performed with SGD, using learning rates $\eta_D = 0.1$, $\eta_G = 0.001$, batch size 1, and up to 1000 iterations (MNIST) or 400 (CIFAR-10).

4. Implementation and Parameter Efficiency

The architecture realizes substantial parameter efficiency. A single VQ-KAN block uses $N_q \times N_d \times N_s$ parameters (e.g., $8 \times 8 \times 4 = 256$), versus $O(10^4)$ for a classical CNN and 48 for an 8-qubit, depth-6 conventional QGAN (without KAN structure). Table 1 provides the parameter counts:

Model	Qubit count/params	Patch size	# Trainable Params
VQ-KAN (GAVQKAN)	$N_q=8, N_d=8, N_s=4$	$16\times16$	256 + spline knots
QGAN	$N_q=8, d=6$	$16\times16$	48
CNN	—	$16\times16$	$O(10^4)$

Measured on MNIST and CIFAR-10, GAVQKAN achieves competitive Sliced Wasserstein Distance and MSE to the reference data using at least an order of magnitude fewer parameters than the classical baseline, and a factor of 5 less than standard CNNs of similar output size (Wakaura, 11 Dec 2025). Training time is 6.5×–11.5× longer than a shallow CNN or QGAN, a consequence of parameter-shift evaluations on quantum hardware/simulators.

5. Experimental Results and Metrics

Evaluations on MNIST (16×16) and CIFAR-10 (downsampled to 22×22) reveal:

• For early training (<400 iterations), GAVQKAN generator loss rises rapidly, then stabilizes; discriminator loss plateaus more gradually, indicating more stable adversarial convergence than CNN or QGAN.
• In Sliced Wasserstein Distance (SWD), GAVQKAN attains the lowest SWD in early epochs for both datasets.
• Generated images become recognizable by iteration ≈100.
• GAVQKAN maintains accuracy and sample diversity in low-data regimes (1000 samples), unlike classical NNs that require larger data and parameter counts.
• A single GAVQKAN generator block of 256 params achieves similar sample realism to much larger classical discriminators.

6. Expressive Power and Theoretical Insights

KANs theoretically approximate any multivariate function using $O(n)$ outer sums rather than $O(n^2)$ parameters of a fully connected network. The quantum extension further boosts expressivity: Born distributions measured from quantum states mix each input's influence non-linearly across all $2^{N_q}$ outcomes, while parametrized spline-based Ry rotations realize highly flexible, nonlinear, high-fidelity feature transformations. GAVQKAN leverages this to compete with classical benchmarks in quality-to-parameter ratio, and the patch-by-patch synthesis allows efficient generation of long output vectors without parameter scaling proportional to data size.

7. Relationship to Prior Quantum-Adversarial and KAN Models

GAVQKAN extends both quantum GAN and KAN frameworks. It differs from conventional quantum GANs (Zoufal et al., 2019, Dallaire-Demers et al., 2018) by the functional role of its generator: KAN-based decomposition is encoded in the variational circuit, allowing for univariate spline-parameterized modules, rather than generic rotation layers. This achieves both parameter efficiency and nonlinear expressivity. Unlike QGANs with simple rotation/entangler ansätze, GAVQKAN implements the full Kolmogorov–Arnold functional mechanism (Wakaura, 11 Dec 2025). Compared to prior hybrid adversarial schemes (Shu et al., 2 Feb 2024, Al-Othni et al., 13 Jul 2025), it takes advantage of KAN's favorable expressivity scaling and quantum evaluation's Born-sampling bottleneck, producing higher accuracy and diversity under data-scarce regimes.

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Key References:

• Wakaura, et al. "Generative Adversarial Variational Quantum Kolmogorov-Arnold Network" (Wakaura, 11 Dec 2025)
• Zoufal, et al. "Quantum Generative Adversarial Networks for Learning and Loading Random Distributions" (Zoufal et al., 2019)
• Shu, et al. "Variational Quantum Circuits Enhanced Generative Adversarial Network" (Shu et al., 2 Feb 2024)