The Amplified Quantum Fourier Transform: solving the local period problem

TL;DR

This paper introduces the Amplified Quantum Fourier Transform (Amplified-QFT), a quantum algorithm that efficiently solves the local period problem by outperforming existing quantum methods in speed.

Contribution

The paper presents the Amplified-QFT algorithm, demonstrating its quadratic speedup over traditional quantum Fourier and hidden subgroup algorithms for the local period problem.

Findings

Amplified-QFT is quadratically faster than QFT and QHS algorithms.

The algorithm effectively finds the local period P with fewer quantum resources.

Detailed proofs and methods for recovering P and s are provided.

Abstract

This paper creates and analyses a new quantum algorithm called the Amplified Quantum Fourier Transform (Amplified-QFT) for solving the following problem: The Local Period Problem: Let L = {0,1...N-1} be a set of N labels and let A be a subset of M labels of period P, i.e. a subset of the form A = {j : j = s + rP; r = 0,1...M-1} where P < sqrt(N) and M << N, and where M is assumed known. Given an oracle f : L->{0,1} which is 1 on A and 0 elsewhere, find the local period P. A separate algorithm finds the offset s. The first part of the paper defines the Amplified-QFT algorithm. The second part of the paper summarizes the main results and compares the Amplified-QFT algorithm against the Quantum Fourier Transform (QFT) and Quantum Hidden Subgroup (QHS) algorithms when solving the local period problem. It is shown that the Amplified-QFT is, on average, quadratically faster than both the QFT and QHS algorithms. The third part of the paper provides the detailed proofs of the main results, describes the method of recovering P from an observation y and describes the method for recovering the offset s.