Quantum Nondecimated Wavelet Transform: Theory, Circuits, and Applications
This work addresses the challenge of integrating classical multiscale signal analysis tools into quantum computing for researchers in quantum signal processing, though it appears incremental as it adapts existing classical methods to a quantum framework.
The paper tackled the problem of embedding the classical nondecimated wavelet transform into quantum computation by developing two quantum formulations that maintain redundancy and translation invariance, resulting in explicit unitary representations and applications like denoising and feature extraction without specific numerical results.
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.