Generalization of Quantum Fourier Transformation

TL;DR

This paper generalizes the quantum Fourier transform over Abelian groups to create more efficient unitaries, explores their relation to the dihedral hidden subgroup problem, and provides explicit formulas for quantum Haar transformation.

Contribution

It introduces a new generalized quantum Fourier transform over Abelian groups and links it to the dihedral hidden subgroup problem, offering explicit formulas for practical use.

Findings

New generalized quantum Fourier transform over Abelian groups

Explicit formulas for the generalized Fourier and Haar transformations

Potential approach to solving the dihedral hidden subgroup problem

Abstract

Quantum Fourier transformation is important in many quantum algorithms. In this paper, we generalize quantum Fourier transformation over the Abelian group $\mathbb{Z}_N$ from two different points to get more efficient unitary transformations. The obtained unitary transformations are given in concise and explicit formula which can be used directly. A relationship between the generalized quantum Fourier transformation and the dihedral hidden subgroup problem are discussed in this paper. This may lead a way to solve the dihedral hidden subgroup problem. The second goal of this paper is to give an explicit formula of quantum Haar transformation.