Properties of long quantum walks in one and two dimensions

Properties of long quantum walks in one and two dimensions The quantum walk (QW) is the term given to a family of algorithms governing the evolution of a discrete quantum system and as such has a founding role in the study of quantum computation. We contribute to the investigation of QW phenomena by performing a detailed numerical study of discrete-time quantum walks. In one dimension (1D), we compute the structure of the probability distribution, which is not a smooth curve but shows oscillatory features on all length scales. By analyzing walks up to N = 1,000,000 steps, we discuss the scaling characteristics and limiting forms of the QW in both real and Fourier space. In 2D, with a view to ready experimental realization, we consider two types of QW, one based on a four-faced coin and the other on sequential flipping of a single two-faced coin. Both QWs may be generated using two two-faced coins, which in the first case are completely unentangled and in the second are maximally entangled. We draw on our 1D results to characterize the properties of both walks, demonstrating maximal speed-up and emerging semi-classical behavior in the maximally entangled QW. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

Properties of long quantum walks in one and two dimensions

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Publisher
Springer US
Copyright
Copyright © 2015 by Springer Science+Business Media New York
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-015-1127-5
Publisher site
See Article on Publisher Site

Abstract

The quantum walk (QW) is the term given to a family of algorithms governing the evolution of a discrete quantum system and as such has a founding role in the study of quantum computation. We contribute to the investigation of QW phenomena by performing a detailed numerical study of discrete-time quantum walks. In one dimension (1D), we compute the structure of the probability distribution, which is not a smooth curve but shows oscillatory features on all length scales. By analyzing walks up to N = 1,000,000 steps, we discuss the scaling characteristics and limiting forms of the QW in both real and Fourier space. In 2D, with a view to ready experimental realization, we consider two types of QW, one based on a four-faced coin and the other on sequential flipping of a single two-faced coin. Both QWs may be generated using two two-faced coins, which in the first case are completely unentangled and in the second are maximally entangled. We draw on our 1D results to characterize the properties of both walks, demonstrating maximal speed-up and emerging semi-classical behavior in the maximally entangled QW.

Journal

Quantum Information ProcessingSpringer Journals

Published: Sep 22, 2015

References

  • An example of the difference between quantum and classical random walks
    Childs, AM; Farhi, E; Gutmann, S
  • Two-dimensional quantum walk with position-dependent phase defects
    Zhang, R; Xue, P
  • A 2d quantum walk simulation of two-particle dynamics
    Schreiber, A; Gábris, A; Rohde, PP; Laiho, K; Stefanak, M; Potoc̈ek, V; Hamilton, C; Jex, I; Silberhorn, C
  • N-dimensional alternate coined quantum walks from a dispersion-relation perspective
    Roldán, E; Franco, C; Silva, F; Valcárcel, GJ

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