The Hidden Subgroup Problem and Eigenvalue Estimation on a Quantum Computer

TL;DR

This paper discusses how the hidden subgroup problem in Abelian groups can be addressed through eigenvalue estimation on quantum computers, highlighting simplified solutions with minimal control qubits.

Contribution

It extends the eigenvalue estimation framework to the general Abelian hidden subgroup problem and introduces solutions using only one control qubit.

Findings

Eigenvalue estimation applies to the hidden subgroup problem

Single control qubit solutions are possible

Enhanced understanding of quantum algorithms for group problems

Abstract

A quantum computer can efficiently find the order of an element in a group, factors of composite integers, discrete logarithms, stabilisers in Abelian groups, and `hidden' or `unknown' subgroups of Abelian groups. It is already known how to phrase the first four problems as the estimation of eigenvalues of certain unitary operators. Here we show how the solution to the more general Abelian `hidden subgroup problem' can also be described and analysed as such. We then point out how certain instances of these problems can be solved with only one control qubit, or `flying qubits', instead of entire registers of control qubits.