Weighted Quantum Signal Processing: Low-Depth Polynomial Approximation with Applications to Kolmogorov-Arnold Networks

Quantum Signal Processing is a powerful quantum framework for generating and approximating univariate polynomials. However, QSP is often limited by circuit-depth bottlenecks and parity constraints on the class of realizable polynomials. In this work, we introduce Weighted Quantum Signal Processing, an extension of QSP in which a weight function is assigned to the central rotation operator. This formulation provides a deeper understanding of QSP, which emerges as the special case of WQSP with unit weights. The choice of weights determines the structure and expressive capabilities of WQSP circuits. When the weights are natural numbers greater than one, WQSP reduces to a pruned version of QSP, revealing parameter redundancies in the standard framework. Through appropriate selection of integer weights, WQSP achieves linear-to-exponential reductions in the number of parameters required to realize arbitrary bounded univariate polynomials while preserving approximation quality. For generic weights, we establish corresponding approximation error bounds and show that, in many cases, the approximation is exact. We analyze WQSP from both a deterministic perspective, where polynomial generation is formulated as the solution of a linear system, and a quantum machine learning perspective, where WQSP serves as a structured and expressive quantum learning model. We further employ this learning framework to parameterize learnable activation functions in Kolmogorov--Arnold Networks for multivariate function approximation. Our results show that WQSP provides a compact, flexible, and theoretically grounded framework for realizing arbitrary univariate polynomials while requiring significantly fewer trainable parameters than conventional QSP. This yields expressive and parameter-efficient neural architectures, highlighting the potential of WQSP as a scalable primitive for quantum-enhanced machine learning.