The dihedral hidden subgroup problem

TL;DR

This paper explores the dihedral hidden subgroup problem, analyzing quantum algorithms' limitations and modifications, and introduces a novel link to quantum state cloning, advancing understanding in quantum algorithm design.

Contribution

It provides a detailed exposition of the dihedral hidden subgroup problem, discusses obstructions and modifications of quantum algorithms, and reveals a new connection to quantum state cloning.

Findings

Standard quantum algorithms face obstructions in solving the problem

Modified algorithms can achieve polynomial quantum query complexity

A new connection between the dihedral coset problem and quantum state cloning is established

Abstract

We give an exposition of the hidden subgroup problem for dihedral groups from the point of view of the standard hidden subgroup quantum algorithm for finite groups. In particular, we recall the obstructions for strong Fourier sampling to succeed, but at the same time, show how the standard algorithm can be modified to establish polynomial quantum query complexity. Finally, we explain a new connection between the dihedral coset problem and cloning of quantum states.