量子非抽样小波变换：理论、电路与应用
摘要
非抽样或平移不变小波变换 (NDWT) 是经典多尺度信号分析中的核心工具，以其稳定性、冗余性和平移不变性著称。本文发展了两种互补的量子 NDWT 形式，将这些经典属性在量子计算中实现为统一的框架。第一种形式基于 NDWT 的 epsilon 抽样解释，通过将平移指数提升为量子寄存器并施加受控平移，同时执行小波分析幺正变换，从而同时实现所有循环平移小波变换。 resulting construction yields an explicit, fully unitary quantum representation of redundant wavelet coefficients and supports coherent postprocessing, including quantum shrinkage via ancilla-driven completely positive trace preserving maps. The second formulation is based on the Hadamard test and uses diagonal phase operators to probe scale-shift wavelet structure through interference, providing direct access to shift-invariant energy scalograms and multiscale spectra without explicit coefficient reconstruction. Together, these two approaches demonstrate that redundancy and translation invariance can be exploited rather than avoided in the quantum setting. Applications to denoising, feature extraction, and spectral scaling illustrate how quantum NDWTs provide a flexible and physically meaningful foundation for multiscale quantum signal processing.
引用
@article{arxiv.2512.21477,
title = {Topological guidance of a self-propelled particle},
author = {Ethan Andersson and Valeri Frumkin},
journal= {arXiv preprint arXiv:2512.21477},
year = {2026}
}