Representation and design of wavelets using unitary circuits

TL;DR

This paper explores representing discrete wavelet transformations as unitary circuits, linking classical wavelet methods with quantum circuit formalism, and introduces algorithms for constructing and designing various wavelet types.

Contribution

It introduces a circuit-based formalism for wavelet transformations, enabling new design approaches and connecting classical and quantum multi-scale methods.

Findings

Explicit circuit representations for Daubechies, coiflets, and symlets wavelets.

Algorithm for constructing circuit representations of known wavelets.

Design of novel wavelets using the circuit formalism.

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 multi-scale representation of quantum many-body wavefunctions using unitary circuits, further cementing the relation established in [Phys. Rev. Lett. 116, 140403 (2016)] between classical and quantum multi-scale 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 novel WTs, including various classes of symmetric wavelets and multi-wavelets, boundary wavelets and biorthogonal wavelets.