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arXiv:1010.0033 · Quantum Information Processing

The amplified quantum Fourier transform: solving the local period problem

David J. Cornwell

Published Aug 14, 2012• 0 citations
arXivHTMLPDF
DOI
10.1007/s11128-012-0468-6

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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 {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.
In this thesis we investigate two new Amplified Quantum Transforms. In particular we create and analyze the Amplified Quantum Fourier Transform (Amplified-QFT) and the Amplified-Haar Wavelet Transform. The Amplified-QFT algorithm is used to solve the Local Period Problem. We calculate the probabilities of success and compare this algorithm with the QFT and QHS algorithms. We also examine the Amplified-QFT algorithm for solving the Local Period Problem with Error Stream. We use the Amplified-Haar Wavelet Transform for solving the Local Constant or Balanced Signal Decision Problem which is a generalization of the Deutsch-Jozsa problem.
In this paper, we introduce quantum fourier transform as a key ingredient for many useful algorithms. These algorithms make a solution for problems which is considered to be intractable problems on a classical computer. Quantum Fourier transform is propounded as a key for quantum phase estimation algorithm. In this paper our aim is the implementation of period-finding algorithm.Quantum computer solves this problem, exponentially faster than classical one. Quantum phase estimation algorithm is the key for the period-finding problem .Therefore, by means of simulating quantum Fourier transform, we are able to implement the period-finding algorithm. In this paper, the simulation of quantum Fourier transform is carried out by Matlab software.