Efficient Quantum Transforms

TL;DR

This paper presents new quantum algorithms and network designs for efficiently computing various unitary transforms, including Fourier and wavelet transforms, with applications to quantum error correction and non-Abelian groups.

Contribution

It introduces novel quantum network constructions for a range of transforms, including generalized Kronecker products and transforms on non-Abelian groups, enhancing quantum algorithm efficiency.

Findings

Networks for Walsh-Hadamard and quantum Fourier transforms re-developed.

New wavelet transform networks for quantum computation introduced.

Efficient quantum networks for metacyclic groups and error-correction group transforms provided.

Abstract

Quantum mechanics requires the operation of quantum computers to be unitary, and thus makes it important to have general techniques for developing fast quantum algorithms for computing unitary transforms. A quantum routine for computing a generalized Kronecker product is given. Applications include re-development of the networks for computing the Walsh-Hadamard and the quantum Fourier transform. New networks for two wavelet transforms are given. Quantum computation of Fourier transforms for non-Abelian groups is defined. A slightly relaxed definition is shown to simplify the analysis and the networks that computes the transforms. Efficient networks for computing such transforms for a class of metacyclic groups are introduced. A novel network for computing a Fourier transform for a group used in quantum error-correction is also given.