Quantum Nondecimated Wavelet Transform: Theory, Circuits, and Applications

TL;DR

This paper introduces two quantum formulations of the nondecimated wavelet transform (NDWT), enabling shift-invariant multiscale analysis in quantum computing with applications in denoising and feature extraction.

Contribution

It develops novel quantum algorithms for NDWT that preserve classical properties like shift invariance and redundancy within quantum computation.

Findings

Quantum NDWT supports coherent postprocessing and shrinkage.

Two formulations enable direct access to multiscale spectra.

Applications include denoising and feature extraction in quantum signals.

Abstract

The nondecimated or translation-invariant wavelet transform (NDWT) is a central tool in classical multiscale signal analysis, valued for its stability, redundancy, and shift invariance. This paper develops two complementary quantum formulations of the NDWT that embed these classical properties coherently into quantum computation. The first formulation is based on the epsilon-decimated interpretation of the NDWT and realizes all circularly shifted wavelet transforms simultaneously by promoting the shift index to a quantum register and applying controlled circular shifts followed by a wavelet analysis unitary. The 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.