Quantum algorithms for shifted subset problems

TL;DR

This paper introduces quantum algorithms for the shifted subset problem in abelian groups, achieving exponential speedups over classical methods by efficiently identifying specific subsets like Hamming spheres.

Contribution

It presents novel quantum algorithms for the shifted subset problem, including Hamming spheres, with polynomial time complexity and exponential advantages over classical algorithms.

Findings

Quantum algorithms efficiently identify subsets in abelian groups.

Exponential separation from classical algorithms demonstrated.

Algorithms applicable to Hamming spheres and boolean cube subsets.

Abstract

We consider a recently proposed generalisation of the abelian hidden subgroup problem: the shifted subset problem. The problem is to determine a subset S of some abelian group, given access to quantum states of the form |S+x>, for some unknown shift x. We give quantum algorithms to find Hamming spheres and other subsets of the boolean cube {0,1}^n. The algorithms have time complexity polynomial in n and give rise to exponential separations from classical computation.