Nondecimated Wavelet Transform Advances Quantum Signal Processing Via Unitary Circuits

The analysis of signals often requires identifying patterns at multiple scales, a process traditionally achieved with wavelet transforms, but these methods can struggle with accurately representing shifts in a signal. Brani Vidakovic from Texas A and M University, along with colleagues, now presents a new quantum approach to non-decimated wavelet transforms that overcomes these limitations. Their work establishes two distinct, yet complementary, methods for performing this analysis using the principles of quantum computation, offering both a complete and unitary representation of wavelet coefficients and a direct way to access scale-shift wavelet structure. This achievement demonstrates that the redundancy inherent in these transforms can be harnessed for improved signal processing, with potential applications in areas like denoising, feature extraction, and spectral scaling, and represents a significant step towards more flexible and physically meaningful multiscale analysis.

Quantum Algorithms for Nondecimated Wavelet Transforms

This paper introduces two quantum algorithms for performing nondecimated wavelet transforms (QNDWT), building upon the classical concept of redundant wavelet analysis. The goal is to leverage quantum computation for efficient multiscale analysis, denoising, and inference. The research demonstrates that NDWT concepts translate well to the quantum realm, offering potential advantages for signal processing and data analysis on future quantum computers. The study centers on the nondecimated wavelet transform (NDWT), a powerful tool for multiscale signal analysis, and pioneers quantum implementations of this technique.

Researchers developed two distinct formulations of the NDWT, both designed for efficient computation within a quantum framework. One approach promotes the shift index to a quantum register, realizing all circularly shifted wavelet transforms simultaneously using controlled circular shifts and a wavelet analysis unitary, creating an explicit coefficient-domain representation with preserved redundancy and shift invariance. Complementing this, the second formulation employs the Hadamard test, utilizing diagonal phase operators to probe the scale-shift wavelet structure through interference, directly accessing shift-invariant energy scalograms and multiscale spectra without explicitly reconstructing coefficients. This innovative method bypasses traditional coefficient reconstruction, offering a streamlined pathway to key signal characteristics.

Both methods replace classical enumeration of shifts with quantum parallelism, preserving the important properties of NDWT, and offer a trade-off between coefficient-level access and direct energy domain access. Scientists developed two new ways to perform NDWTs within quantum computation, maintaining the stability, redundancy, and shift invariance that make NDWTs valuable while adapting them for quantum systems. Experiments demonstrate that these transforms can be implemented coherently, allowing for post-processing including shrinkage via ancilla-driven completely positive trace preserving maps. The team realized all circularly shifted wavelet transforms simultaneously by treating the shift index as a register and applying controlled circular shifts before performing a standard wavelet analysis, yielding explicit, fully unitary representations.

Measurements confirm that the epsilon-decimated family, consisting of 2L ordinary wavelet transforms, produces a total of N2L coefficients for a signal of length N, where L represents the multiresolution depth. The team validated these methods with numerical examples, implemented using qiskit and publicly available, demonstrating the flexibility and physical meaningfulness of NDWTs for multiscale signal processing. These methods offer potential for applications in multiscale analysis, denoising, and inference, with future research focused on optimized compilation of wavelet unitaries, hybrid algorithms, and extension to more complex signals. This research presents two new ways to perform NDWTs using the principles of quantum computation, effectively embedding classical multiscale signal analysis into a quantum framework. The team successfully demonstrated how to represent redundant wavelet information coherently, meaning the transform can process all possible shifted versions of a signal simultaneously, and how to exploit this redundancy for improved signal processing. The researchers highlight the potential to balance computational cost, coherence, and robustness, adapting the method to suit specific hardware limitations and application needs.