The central nature of the Hidden Subgroup problem

TL;DR

This paper demonstrates the fundamental role of the Hidden Subgroup problem in quantum computing by establishing equivalences among various related problems and providing new algorithmic tools for their analysis.

Contribution

It shows the equivalence and reducibility of several quantum problems to Hidden Subgroup across different groups, and introduces nonadaptive checkers for these problems.

Findings

Hidden Coset, Hidden Shift, and Orbit Coset are equivalent or reducible to Hidden Subgroup.

Over permutation groups, decision and search versions of Hidden Subgroup are polynomial-time equivalent.

Nonadaptive program checkers for Hidden Subgroup and its decision version are developed.

Abstract

We show that several problems that figure prominently in quantum computing, including Hidden Coset, Hidden Shift, and Orbit Coset, are equivalent or reducible to Hidden Subgroup for a large variety of groups. We also show that, over permutation groups, the decision version and search version of Hidden Subgroup are polynomial-time equivalent. For Hidden Subgroup over dihedral groups, such an equivalence can be obtained if the order of the group is smooth. Finally, we give nonadaptive program checkers for Hidden Subgroup and its decision version.