Variational Quantum Kolmogorov-Arnold Network

• VQKAN is a hybrid quantum-classical framework that leverages the Kolmogorov-Arnold theorem to decompose continuous multivariate functions using learnable quantum circuits.
• It employs advanced quantum techniques like Quantum Signal Processing, block encoding, and single-qubit data re-uploading to realize adaptive activation functions.
• VQKAN demonstrates robust performance in quantum state preparation, chemistry, and optimization, reducing parameters while enhancing efficiency for NISQ devices.

A Variational Quantum Kolmogorov-Arnold Network (VQKAN) is a class of hybrid quantum-classical algorithms that integrate the universal function decomposition properties of Kolmogorov-Arnold Networks (KANs) with the expressive power and optimization routines of variational quantum circuits. VQKANs are designed to model complex multivariate mappings by leveraging the Kolmogorov-Arnold representation theorem, exploiting quantum resources to encode, process, and optimize functions efficiently in various machine learning, quantum simulation, and combinatorial optimization tasks. Unlike standard quantum neural networks (QNNs) that employ fixed activation functions and fixed circuit architectures, VQKANs use learnable functional transformations along network edges, often realized via parameterized quantum gates or circuits which are optimized to minimize user-defined cost functions, such as ground state energy or classification loss.

1. Mathematical Foundations and Core Principles

VQKANs are constructed around the Kolmogorov-Arnold representation theorem, which asserts that any continuous multivariate function $f(x_1, ..., x_n)$ can be decomposed as a finite sum of compositions of univariate functions:

$$f(x_1,\dots,x_n) = \sum_{q=0}^{2n} \Phi_q \left[ \sum_{p=1}^n \phi_{q,p}(x_p) \right].$$

Classical KANs instantiate the functions $\phi_{q,p}$ and $\Phi_q$ as spline-based or basis-function parameterizations, with training performed across network edges rather than nodes. In VQKANs, these functional transformations are encoded in parameterized quantum circuits, where variational parameters control the gate operations replacing scalar weights. Quantum architectures thus adopt a neuromorphic structure, retaining interpretability and expressive capacity while facilitating the quantum parallelism and entanglement properties necessary for deep quantum learning (Kundu et al., 25 Jun 2024).

2. Quantum Circuit Implementations and Variational Ansatz

The quantum implementation of KANs can proceed by multiple routes, primarily:

• Quantum Signal Processing (QSP): Activation functions are realized as polynomial approximations using QSP circuits,

$$U_{\vec{\phi}}(a) = e^{i\phi_0 \sigma_z} \prod_{k=1}^d [W(a) e^{i\phi_k \sigma_z}],$$

where $W(a)$ is a parameterized rotation (e.g., $$W(a) = \begin{pmatrix}a& i\sqrt{1-a^2}\ i\sqrt{1-a^2}& a\end{pmatrix}$$), and the phases $\vec{\phi}$ are variationally optimized (Daskin, 5 Oct 2024).

• Block-Encoding and Quantum Singular Value Transformation (QSVT): Input data and trainable parameters are encoded as diagonal blocks in unitary operators, enabling efficient polynomial activation implementation via QSVT. For example, Chebyshev polynomial-based activations are realized as:

$$\phi_{pq}(x) = \frac{1}{d+1} \sum_{r=0}^{d} w_{pq}^{(r)} T_r(x),$$

with $T_r(x)$ a Chebyshev polynomial, weights $w_{pq}^{(r)}$ as parameters, and the overall gate complexity scaling as $O(d^{2L}/2^L)$ for $L$ layers (Ivashkov et al., 6 Oct 2024).

• Variational Ansatz with Adaptive Tiling: Recent enhancements for NISQ devices propose variational circuits designed via tiling, where layer matrices are recursively partitioned and emulated with block-diagonal unitary operators, reducing parameter count to $\sim 2^{N_q-1}$ for $N_q$ qubits and eliminating the need for QSP or heavy block encoding (Wakaura et al., 28 Mar 2025).
• Single-Qubit Data Re-Uploading Circuits (DARUAN): Quantum variational activation functions (QVAFs) are realized using single-qubit data re-uploading techniques. Sequential data encoding gates $S(w_\ell x) = e^{-i w_\ell x H}$, repeated $r$ times, expand the frequency spectrum exponentially with depth (maximal frequency $K_B = 2^r-1$), yielding activation functions as quantum expectation values:

$$\phi_{\bm{\theta}}(x) = \langle 0| U(x; \bm{\theta})^\dagger \mathcal{M} U(x;\bm{\theta}) |0\rangle,$$

where $U$ is the variational circuit, and $\mathcal{M}$ is a chosen observable (Jiang et al., 17 Sep 2025).

3. Optimization and Adaptive Methods

Parameter optimization in VQKANs proceeds via hybrid quantum-classical loops, utilizing cost functions dictated by the target application (e.g., energy minimization for quantum chemistry, classification loss, combinatorial objective for TSP) (Wakaura, 26 Sep 2025). Advances in training include:

• Adaptive Ansatz Construction: VQKANs may employ adaptive ansatz strategies, akin to Adaptive VQE, where circuit complexity grows incrementally. At each iteration, candidate gates/operators are appended from a pool only if they yield improvements in the loss function. This avoids overparameterization and ensures efficient convergence (Wakaura et al., 27 Mar 2025).
• Quantum Imaginary Time Evolution (QITE) Optimization: QITE can be used to update parameters by emulating damping towards the ground state:

$$\delta\theta_j = -\Delta\tau \sum_k S_{jk}^{-1} \; \text{Re}\langle \partial_{\theta_k} \psi | H | \psi\rangle,$$

where $S_{jk}$ is the quantum geometric tensor, $H$ is the objective Hamiltonian, and the update ensures monotonic energy minimization (Wakaura et al., 28 Jun 2025).

• Hybrid and Fully Quantum Architectures: The network may delegate function computation either to hybrid quantum-classical models (with classical post-processing or activation) or fully quantum models (where function evaluation and activation occur entirely within quantum amplitudes, e.g., QCBM with amplitude encoding) (Werner et al., 27 Jun 2025).

4. Applications in Quantum State Preparation, Chemistry, and Optimization

VQKANs have demonstrated efficacy in diverse domains:

• Quantum State Preparation: In tasks such as Bell and GHZ state generation, VQKAN-based RL agents outperform standard MLP-based designs, achieving probabilities of success $2$–$5\times$ higher for optimal admissible circuits and greater fidelity under noise (e.g., $73.28\%$ fidelity for noisy environments) (Kundu et al., 25 Jun 2024).
• Quantum Chemistry and VQE: Curriculum RL integration with VQKAN enables efficient ground state searches for molecules (H$_2$, LiH), reducing 2-qubit gate counts and circuit depth, with parameter efficiency (e.g., $5.525\times10^4$ parameters vs. $5.15\times10^6$ for MLPs) and robust convergence towards chemical accuracy (e.g., within $0.0016$ Hartree) (Kundu et al., 25 Jun 2024).
• Combinatorial Optimization (e.g., TSP): VQKAN provides a route to encode combinatorial paths using only $N$ qubits for $N$ sites, leveraging parameterized rotations and swap operations. A loss function with taboo terms penalizes invalid routes (e.g., repetition), and hybrid optimization schemes outperform conventional VQE, adjusting dynamically to time-dependent edge weights (Wakaura, 26 Sep 2025).
• Quantum Many-Body Wavefunctions: Kolmogorov-Arnold ansatz variants (e.g., SineKAN, spline-based KAN) have been used in variational Monte Carlo, achieving competitive or superior representational power with order-of-magnitude lower computational cost compared to standard MLPs, and can be extended with explicit two-body "cusp" terms to accurately capture short-range interactions (Shamim et al., 2 Jun 2025, Bedaque et al., 2 Jun 2025).

5. Interpretability, Scalability, and Resource Requirements

A salient feature of VQKANs is their interpretability: network edge functions are explicit, inspectable, and often parameterized by splines or polynomial expansions. Compared to classical MLPs, VQKANs frequently require $2$–$3\times$ fewer trainable parameters, but modestly higher episode execution time ($2.8$–$4\times$ increase). Recent innovations (MultKAN, tiling, hybrid bottleneck architectures) have further reduced computation overhead (Kundu et al., 25 Jun 2024, Wakaura et al., 28 Mar 2025, Jiang et al., 17 Sep 2025).

Scalability is addressed through layer extension (incremental deepening of the variational activation block), hybrid architectures (HQKAN as MLP replacement), and coarse-to-fine spline refinement, ensuring the model's tractability for high-dimensional and large-scale tasks. Gate complexity generally grows exponentially with layer count in block-encoded models ($O(d^{2L}/2^L)$ for $L$ layers and degree $d$), but can be mitigated in NISQ-optimized tiling or single-qubit re-uploading designs (Ivashkov et al., 6 Oct 2024, Jiang et al., 17 Sep 2025, Wakaura et al., 28 Mar 2025).

6. Experimental Performance and Physical Constraints

Benchmarks confirm VQKAN’s effectiveness:

• In regression and generative modeling, QKAN-based modules yielded lower RMSE and perplexity scores compared to both MLPs and classical KANs, benefiting from exponential parameter reduction via frequency spectrum expansion in DARUANs (Jiang et al., 17 Sep 2025).
• In image classification, VQKAN architectures matched or surpassed baseline models on standard datasets (MNIST, CIFAR-10/100), achieving similar accuracy with $30\%$–$70\%$ fewer parameters in dense layers (Jiang et al., 17 Sep 2025).
• In quantum time series analysis, physics-informed KANs with Ehrenfest-constrained loss functions achieved accurate dynamical predictions with $5.4\%$ of the data required by Temporal Convolutional Networks, maintaining causal consistency via the Chain of KANs architecture (Sen et al., 23 Sep 2025).

7. Future Directions and Extensions

Expansion of VQKAN frameworks is directed towards:

• Integration of block encoding and tensor network methods for deeper architectures and higher-dimensional problems.
• Advanced optimization routines to mitigate barren plateaus in training.
• Realization and benchmarking of VQKAN and EVQKAN circuits on actual NISQ devices, focusing on noise robustness and error mitigation (Wakaura et al., 28 Mar 2025).
• Hybrid quantum-classical deployments, including knowledge distillation workflows where quantum-trained modules inform classical networks, and extension to large-scale models such as LLMs through HQKAN replacements.
• Applications in scheduling, logistics, network optimization, and quantum simulation, exploiting qubit-efficient encodings and functional decompositions (Wakaura, 26 Sep 2025).

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VQKAN encapsulates a theoretically rigorous and practically promising paradigm for quantum-enhanced machine learning, variational optimization, and quantum circuit design, blending the universality and interpretability of Kolmogorov-Arnold architectures with the adaptability and computational advantages of quantum circuits. As demonstrated by multi-domain experiments, adaptive strategies, resource-efficient designs, and physical-constraint embedding, VQKAN sets a benchmark for next-generation quantum algorithms, methodologies, and applications.