The Hidden Subgroup Problem - Review and Open Problems

TL;DR

This paper reviews the Hidden Subgroup Problem in quantum computing, providing detailed proofs, unified notation, and outlining open problems, especially focusing on the abelian case and future directions for nonabelian groups.

Contribution

It offers a comprehensive mathematical overview with detailed proofs and algorithms for the abelian case, and highlights open problems and future research directions.

Findings

Concrete algorithms and bounds for finite abelian groups

Unified notation simplifies understanding of the problem

Identification of open problems in nonabelian cases

Abstract

An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint. Detailed proofs are supplied for many important results from the literature, and notation is unified, making it easier to absorb the background necessary to begin research on the Hidden Subgroup Problem. Proofs are provided which give very concrete algorithms and bounds for the finite abelian case with little outside references, and future directions are provided for the nonabelian case. This summary is current as of October 2004.