Quantum wavelet transforms of any order

Abstract

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.