加权量子信号处理：低深度多项式逼近及其在Kolmogorov-Arnold网络中的应用
Weighted Quantum Signal Processing: Low-Depth Polynomial Approximation with Applications to Kolmogorov-Arnold Networks
- Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
- Universidad de Oviedo（奥维耶多大学）
- CERN（欧洲核子研究中心）
- Centre for Quantum Technologies (CQT), National University of Singapore（新加坡国立大学量子技术中心）
- Universitat Politècnica de València（瓦伦西亚理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出加权量子信号处理（WQSP），通过为旋转算子分配权重扩展QSP，实现低深度多项式逼近，并用于Kolmogorov-Arnold网络的可学习激活函数，显著减少可训练参数。
AI中文摘要：
量子信号处理（QSP）是一个强大的量子框架，用于生成和逼近单变量多项式。然而，QSP常常受到电路深度瓶颈和可实现多项式类别的奇偶性约束的限制。在这项工作中，我们引入了加权量子信号处理（WQSP），这是QSP的一种扩展，其中权重函数被分配给中心旋转算子。这一表述提供了对QSP更深入的理解，QSP作为单位权重的WQSP特例出现。权重的选择决定了WQSP电路的结构和表达能力。当权重为大于1的自然数时，WQSP简化为QSP的剪枝版本，揭示了标准框架中的参数冗余。通过适当选择整数权重，WQSP在保持逼近质量的同时，实现任意有界单变量多项式所需参数数量从线性到指数的减少。对于一般权重，我们建立了相应的逼近误差界，并表明在许多情况下逼近是精确的。我们从确定性视角（多项式生成被表述为线性系统的解）和量子机器学习视角（WQSP作为结构化且富有表达力的量子学习模型）分析WQSP。我们进一步利用这一学习框架参数化Kolmogorov-Arnold网络中的可学习激活函数，用于多元函数逼近。我们的结果表明，WQSP提供了一个紧凑、灵活且具有理论基础的框架，用于实现任意单变量多项式，同时所需的可训练参数显著少于传统QSP。这产生了富有表达力和参数高效的神经架构，突显了WQSP作为量子增强机器学习可扩展原语的潜力。
英文摘要：
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.
