Quantum Algorithms for some Hidden Shift Problems

TL;DR

This paper introduces quantum algorithms for solving hidden shift problems using the quantum Fourier transform, expanding the understanding of shift and subgroup structures in quantum computation.

Contribution

It presents three new hidden shift problems solvable efficiently with quantum Fourier transform and defines the hidden coset problem as a unifying framework.

Findings

Quantum algorithms solve hidden shift problems efficiently.

The hidden coset problem generalizes existing problems.

Fourier transform captures shift and subgroup structures.

Abstract

Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structure of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of ``unknown shift'' problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.