Enhanced Variational Quantum Kolmogorov-Arnold Network

TL;DR

The paper introduces EVQKAN, a tiling-based variational quantum circuit that emulates Kolmogorov-Arnold Network layers without relying on Quantum Signal Processing, addressing the mismatch between KAN and current NISQ hardware. EVQKAN demonstrates higher accuracy than VQKAN and QNN on fitting elementary functions and 2D classification, even with a single layer, though classification results reveal variability and overfitting in some setups. The approach leverages sum-operator tiling, spline-like parameterizations, and a VQE-style objective to enable robust learning on near-term devices. While promising for practical quantum ML, the method faces exponential scaling in runtime with deeper circuits, motivating circuit-optimization, block encoding, and resilience benchmarking as key next steps.

Abstract

The Kolmogorov-Arnold Network (KAN) is a novel multi-layer network model recognized for its efficiency in neuromorphic computing, where synapses between neurons are trained linearly. Computations in KAN are performed by generating a polynomial vector from the state vector and layer-wise trained synapses, enabling efficient processing. While KAN can be implemented on quantum computers using block encoding and Quantum Signal Processing, these methods require fault-tolerant quantum devices, making them impractical for current Noisy Intermediate-Scale Quantum (NISQ) hardware. We propose the Enhanced Variational Quantum Kolmogorov-Arnold Network (EVQKAN) to overcome this limitation, which emulates KAN through variational quantum algorithms. The EVQKAN ansatz employs a tiling technique to emulate layer matrices, leading to significantly higher accuracy compared to conventional Variational Quantum Kolmogorov-Arnold Network (VQKAN) and Quantum Neural Networks (QNN), even with a smaller number of layers. EVQKAN achieves superior performance with a single-layer architecture, whereas QNN and VQKAN typically struggle. Additionally, EVQKAN eliminates the need for Quantum Signal Processing, enhancing its robustness to noise and making it well-suited for practical deployment on NISQ-era quantum devices.

Figures (11)

• Figure 1: Simplified picture of our ansatz on EVQKAN. White circle indicates that connected operators are acted on the circuit in case the qubits that the circle exists is $\mid 0 \rangle$ state and black circle indicates that connected operators are acted on the circuit in case the qubits that the circle exists is $\mid 1 \rangle$ state, respectively.
• Figure 2: Ansatz structure of the Quantum Neural Network (QNN) with parameterized rotation gates controlled by trainable parameters $\theta_j$. Inputs $x_i$ encode classical data into the quantum circuit, enabling learning through optimization of $\theta_j$ to minimize the cost function.
• Figure 3: Number of trials vs. loss functions for optimization of the fitting problem by ( a ) QNN, ( b ) VQKAN, ( c ) Adaptive VQKAN, and ( d ) Enhanced VQKAN, respectively.
• Figure 4: (Right) Number of test points vs. average and median of loss functions (absolute distances) in log10 scale of test points on ( blue line ) QNN, ( green line ) VQKAN, ( orange line ) Adaptive VQKAN, and ( red line ) Enhanced VQKAN optimization, respectively. The line of QNN, VQKAN, and Adaptive VQKAN, are moved right 0.375, 0.25, and 0.125, respectively.
• Figure 5: (Left) Number of trials vs. loss functions for attempts on classification problem using QNN. (Right) Number of test points vs. the average and median loss functions (absolute distances) of test points using QNN optimization.