Quantum Variational Activation Functions Empower Kolmogorov-Arnold
  Networks
Authors: name: Jiun-Cheng Jiang, name: Morris Yu-Chao Huang, name: Tianlong Chen, name: Hsi-Sheng Goan

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
quantum variational activation functions (QVAFs), realized through single-qubit
data re-uploading circuits called DatA Re-Uploading ActivatioNs (DARUANs). We
show that DARUAN with trainable weights in data pre-processing possesses an
exponentially growing frequency spectrum with data repetitions, enabling an
exponential reduction in parameter size compared with Fourier-based activations
without loss of expressivity. Embedding DARUAN into KANs yields
quantum-inspired KANs (QKANs), which retain the interpretability of KANs while
improving their parameter efficiency, expressivity, and generalization. We
further introduce two novel techniques to enhance scalability, feasibility and
computational efficiency, such as layer extension and hybrid QKANs (HQKANs) as
drop-in replacements of multi-layer perceptrons (MLPs) for feed-forward
networks 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. DARUANs and QKANs offer a promising direction for advancing quantum
machine learning on both noisy intermediate-scale quantum (NISQ) hardware and
classical quantum simulators.