Quantum Nondecimated Wavelet Transform: Theory, Circuits, and Applications

2025-12-24

AI Summary

Quantum computing lacks translation-invariant, redundant, and stable multi-scale transforms—critical for robust signal analysis. Method: We propose the first two quantum non-decimated wavelet transform (Q-NDWT) schemes. Leveraging an ε-downsampling interpretation and a Hadamard-test-based dual-path framework, we represent redundant wavelet coefficients via fully unitary operations. Translation invariance is achieved through a displacement-index register, controlled cyclic shifts, wavelet-analysis unitaries, and ancilla-assisted CPTP mappings—enabling direct energy spectrum measurement without explicit reconstruction. Contribution/Results: (1) First rigorous quantum realization of core classical NDWT properties—translation invariance, redundancy, and stability—within a physically realizable quantum circuit model; (2) Enables coherent post-processing and quantum shrinkage, significantly enhancing noise resilience, multi-scale spectral analysis, and fractal feature extraction in terms of both physical feasibility and estimation accuracy.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Data Mining & Knowledge Management: Data Compression

Application Category

• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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.

Problem

• Develops quantum formulations of translation-invariant wavelet transforms
• Enables coherent multiscale signal analysis in quantum computation
• Applies quantum wavelet transforms to denoising and feature extraction

Innovation

• Quantum NDWT via controlled circular shifts and wavelet unitaries
• Hadamard test with diagonal phase operators for scalograms
• Unitary quantum representation enabling coherent postprocessing and shrinkage