Polynomial-time quantum algorithm for solving the hidden subgroup problem

TL;DR

This paper presents a polynomial-time quantum algorithm that efficiently solves both Abelian and non-Abelian hidden subgroup problems by reducing them to a structured search problem and applying multistep quantum computation.

Contribution

The authors introduce a novel quantum algorithm that reduces the hidden subgroup problem to a structured search, enabling polynomial-time solutions for both Abelian and non-Abelian cases.

Findings

Efficient polynomial-time quantum algorithm for HSP

Reduces HSP to a nested structured search problem

Solves both Abelian and non-Abelian HSP in polynomial time

Abstract

The hidden subgroup problem~(HSP) is one of the most important problems in quantum computation. Many problems for which quantum algorithm achieves exponential speedup over its classical counterparts can be reduced to the Abelian HSP. However, there is no efficient quantum algorithm for solving the non-Abelian HSP. We find that the HSP can be reduced to a nested structured search problem that is solved efficiently by using a quantum algorithm via multistep quantum computation. Then we solve the HSP and problems that can be reduced to both the Abelian and the non-Abelian HSP in polynomial time by using this algorithm.