Quantum Wavelet Transforms: Fast Algorithms and Complete Circuits

Abstract

Abstract. The quantum Fourier transform (QFT), a quantum analog of the classical Fourier transform, has been shown to be a powerful tool in developing quantum algorithms. However, in classical computing there is another class of unitary transforms, the wavelet transforms, which are every bit as useful as the Fourier transform. Wavelet transforms are used to expose the multi-scale structure of a signal and are likely to be useful for quantum image processing and quantum data compression. In this paper, we derive efficient, complete, quantum circuits for two representative quantum wavelet transforms, the quantum Haar and quantum Daubechies D (4) transforms. Our approach is to factor the classical operators for these transforms into direct sums, direct products and dot products of unitary matrices. In so doing, we find that permutation matrices, a particular class of unitary matrices, play a pivotal role. Surprisingly, we find that operations that are easy and inexpensive to implement classically are not always easy and inexpensive to implement quantum mechanically, and vice versa. In particular, the computational cost of performing certain permutation matrices is ignored classically because they can be avoided explicitly. However, quantum mechanically, these permutation operations must be performed explicitly and hence their cost enters into the full complexity measure of the quantum transform. We consider the particular set of permutation matrices arising in quantum wavelet transforms and develop efficient quantum circuits that implement them. This allows us to design efficient, complete quantum circuits for the quantum wavelet transform.

Unitary Matrix Functions, Wavelet Algorithms, and Structural Properties of Wavelets

“…Hence, the standard formulas from [Wic93], [Kla99], and [FiWi99] for the quantum computing algorithm naturally generalize to the case N > 2 via (2.2.8). Instead of k-registers…”

New Techniques for Daubechies Wavelets and Multiwavelets Implementation using Quantum Computing

“…In this paper, efficient and complete quantum circuits are derived for the quantum Daubechies D (4) wavelet and multiwavelet transform representations. Quantum algorithms are describable in terms of unitary transformations.…”

“…Fortunately, there is an important class of computations, the unitary transforms, such as the Fourier transform, WalshHadamard transform and wavelet transforms, that are describable in terms of unitary operators [4]. The Fourier and Walsh-Hadamard transforms have been used most extensively by the quantum computing algorithms [5] - [11].…”

“…The main issue in deriving feasible and efficient quantum circuits for the quantum wavelet transforms considered in this paper, is the design of efficient quantum circuits for certain permutation matrices. The permutation matrices, due to their specific structure, represent a very special subclass of unitary matrices [4]. Therefore, the exploitation of this specific structure represents the key to achieve an efficient quantum implementation of permutation matrices.…”

“…Therefore, the exploitation of this specific structure represents the key to achieve an efficient quantum implementation of permutation matrices. In this paper, two representative wavelet kernels are considered, the Daubechies D (4) wavelets and multiwavelets. Two new decompositions which lead to gate-level circuits for their implementations are developed.…”