On solving systems of random linear disequations

TL;DR

This paper presents a polynomial-time algorithm for solving systems of random linear disequations in finite abelian p-groups, which is a key step towards quantum algorithms for the hidden subgroup problem.

Contribution

It introduces an efficient algorithm for solving systems of random disequations in abelian p-groups, advancing quantum hidden subgroup problem solutions.

Findings

Algorithm runs in polynomial time in N

Applicable to abelian p-groups with exponent q

Supports quantum hidden subgroup algorithms

Abstract

An important subcase of the hidden subgroup problem is equivalent to the shift problem over abelian groups. An efficient solution to the latter problem would serve as a building block of quantum hidden subgroup algorithms over solvable groups. The main idea of a promising approach to the shift problem is reduction to solving systems of certain random disequations in finite abelian groups. The random disequations are actually generalizations of linear functions distributed nearly uniformly over those not containing a specific group element in the kernel. In this paper we give an algorithm which finds the solutions of a system of N random linear disequations in an abelian p-group A in time polynomial in N, where N=(log|A|)^{O(q)}, and q is the exponent of A.