The Quantum Fourier Transform and Extensions of the Abelian Hidden Subgroup Problem

TL;DR

This paper improves quantum Fourier transform circuits, extends abelian hidden subgroup problem solutions, and explores period-finding over real numbers, revealing new quantum algorithm capabilities and complexity insights.

Contribution

It introduces improved parallel circuits for the QFT, extends solutions to broader hidden subgroup problems, and analyzes the complexity of period-finding over reals.

Findings

Enhanced quantum Fourier transform circuits for cyclic groups

Extended hidden subgroup problem solutions to finitely-generated abelian groups

Demonstrated period-finding over reals is computationally harder than over integers

Abstract

The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the abelian hidden subgroup problem, of which Shor's celebrated factoring and discrete log algorithms are a special case. We begin by addressing various computational issues surrounding the QFT and give improved parallel circuits for both the QFT over a power of 2 and the QFT over an arbitrary cyclic group. These circuits are based on new insight into the relationship between the discrete Fourier transform over different cyclic groups. We then exploit this insight to extend the class of hidden subgroup problems with efficient quantum solutions. First we relax the condition that the underlying hidden subgroup function be distinct on distinct cosets of the subgroup in question and show that this relaxation can be solved whenever G is a finitely-generated abelian group. We then extend this reasoning to the hidden cyclic subgroup problem over the reals, showing how to efficiently generate the bits of the period of any sufficiently piecewise-continuous function on R. Finally, we show that this problem of period-finding over R, viewed as an oracle promise problem, is strictly harder than its integral counterpart. In particular, period-finding over R lies outside the complexity class MA, a class which contains period-finding over the integers.