Quantum Nondecimated Wavelet Transform: Theory, Circuits, and Applications

TL;DR

This work presents two quantum formulations of the nondecimated wavelet transform (NDWT) that preserve translation invariance and redundancy in a quantum setting. The epsilon-decimated approach embeds all circular shifts into a coherent superposition using a shift register and a single wavelet unitary, enabling coefficient-domain postprocessing and quantum shrinkage via CPTP maps. The Hadamard-test formulation provides direct access to shift-invariant energy summaries, producing scalograms and spectra without explicit coefficient reconstruction and is particularly suited to near-term devices. Together, these frameworks unify multiscale, translation-invariant analysis within quantum circuits and offer flexible pathways for denoising, feature extraction, and spectral analysis in quantum signal processing.

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.

Figures (10)

• Figure 1: Quantum epsilon–decimation architecture. An $L$–qubit ancilla register is placed in a uniform superposition by $H^{\otimes L}$. Conditioned on $\varepsilon$, the data register undergoes $S^\varepsilon$ followed by the wavelet analysis unitary $W$. Each coherent branch implements one epsilon–decimated wavelet transform.
• Figure 2: Fully quantum two–level Haar QNDWT on $N=8$. The circuit coherently prepares four shifted Haar transforms in superposition.
• Figure 3: (Left) Doppler test signal sampled at $N=64$ equispaced (in $t$) points. (Right) Quantum Nondecimated Wavelet Transform of depth $L=3.$
• Figure 4: Hadamard test circuit for a nondecimated wavelet at scale $j$ and shift $k$. The data register holds the amplitude-encoded signal $|y\rangle$. The ancilla controls the diagonal phase operator $U_{\phi,j,k}$ and is measured in the $Z$ basis to estimate $\mathbb{E}[Z_{j,k}] = \operatorname{Re}\langle y|U_{\phi,j,k}|y\rangle$. To access the imaginary part, an $S$ gate is inserted on the ancilla before the final Hadamard.
• Figure 5: (Left) $W_{\small QNDWT}$ of the Doppler function sampled at $N = 128$ equispaced points; (Right) The Hadamard estimate for $|w_k|$ agrees with the classical energy up to numerical precision

Theorems & Definitions (4)

• Example 1
• Example 2
• Example 3
• Example 4