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arXiv:1605.07312 · Physical Review A

Representation and design of wavelets using unitary circuits

Glen Evenbly, Steven R. White

Published May 11, 2018• 28 citations
arXivHTMLPDF

Abstract

The representation of discrete, compact wavelet transformations (WTs) as circuits of local unitary gates is discussed. We employ a similar formalism as used in the multiscale representation of quantum many-body wave functions using unitary circuits, further cementing the relation established in the literature between classical and quantum multiscale methods. An algorithm for constructing the circuit representation of known orthogonal, dyadic, discrete WTs is presented, and the explicit representation for Daubechies wavelets, coiflets, and symlets is provided. Furthermore, we demonstrate the usefulness of the circuit formalism in designing WTs, including various classes of symmetric wavelets and multiwavelets, boundary wavelets, and biorthogonal wavelets.

DOI
10.1103/physreva.97.052314

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The representation of discrete, compact wavelet transformations (WTs) as circuits of local unitary gates is discussed. We employ a similar formalism as used in the multiscale representation of quantum many-body wave functions using unitary circuits, further cementing the relation established in the literature between classical and quantum multiscale methods. An algorithm for constructing the circuit representation of known orthogonal, dyadic, discrete WTs is presented, and the explicit representation for Daubechies wavelets, coiflets, and symlets is provided. Furthermore, we demonstrate the usefulness of the circuit formalism in designing WTs, including various classes of symmetric wavelets and multiwavelets, boundary wavelets, and biorthogonal wavelets.
In this paper, new techniques to implement the Daubechies wavelets and multiwavelets are presented using quantum computing synthesis structures.Also, a new quantum implementation of inverse Daubechies multiwavelet transform is proposed.The permutation matrices, particular unitary matrices, play a pivotal role.The particular set of permutation matrices arising in quantum wavelet and multiwavelet transforms is considered, and efficient quantum circuits that implement them are developed.This allows the design of efficient and complete quantum circuits for the quantum wavelet and multiwavelet transforms.
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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.
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Many classical algorithms are known to efficiently compute the wavelet transforms. However, those classical algorithms cannot be directly translated to quantum algorithms. Recently, efficient and complete quantum algorithms for two representative wavelet transforms (quantum Haar and quantum Daubechies of fourth order) have been proposed. In this paper, we generalize these algorithms in order to they can be applied to Daubechies wavelet kernels of any order. Specifically, we develop a method that efficiently factorize those kernels. The factorization is compatible with the existing pyramidal and packet quantum wavelet algorithms. All steps of the algorithm are unitary and easily implementable on a quantum computer.