A new quantum algorithm for the hidden shift problem in $\mathbb{Z}_{2^t}^n$

TL;DR

This paper introduces a quantum algorithm for the hidden shift problem in groups of the form Z_k^n, specifically when k is a power of 2, achieving polynomial time and efficient space usage, with potential applications in broader hidden subgroup problems.

Contribution

The paper presents the first polynomial-time quantum algorithm for the hidden shift problem in Z_{2^t}^n, improving efficiency for this class of groups.

Findings

Polynomial running time in n for k as a power of 2

Quadratic classical and linear quantum space complexity

Potential to enhance algorithms for hidden subgroup problems

Abstract

In this paper we make a step towards a time and space efficient algorithm for the hidden shift problem for groups of the form $\mathbb{Z}_k^n$. We give a solution to the case when $k$ is a power of 2, which has polynomial running time in $n$, and only uses quadratic classical, and linear quantum space in $n\log (k)$. It can be a useful tool in the general case of the hidden shift and hidden subgroup problems too, since one of the main algorithms made to solve them can use this algorithm as a subroutine in its recursive steps, making it more efficient in some instances.