A Quantum Polynomial-Time Solution to The Dihedral Hidden Subgroup Problem

TL;DR

This paper introduces a polynomial-time quantum algorithm for solving the Dihedral Hidden Subgroup Problem, significantly improving over previous exponential-time algorithms by leveraging the problem's structure.

Contribution

The paper presents the first polynomial-time quantum algorithm for the dihedral hidden subgroup problem, advancing quantum algorithms for non-abelian groups.

Findings

Achieved polynomial-time complexity for the problem

Utilized subgroup lattice structure to guide the quantum walk

Improved over previous exponential-time algorithms

Abstract

We present a polynomial-time quantum algorithm for the Hidden Subgroup Problem over $\mathbb{D}_{2^n}$. The usual approach to the Hidden Subgroup Problem relies on harmonic analysis in the domain of the problem, and the best known algorithm using this approach has time complexity in $2^{\mathcal{O}(\sqrt{n})}$. By focusing on structure encoded in the codomain of the problem, we develop a polynomial-time algorithm which uses this structure to direct a "walk" down the subgroup lattice of $\mathbb{D}_{2^n}$ terminating at the hidden subgroup.