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Related Experiment Videos

Weak limits for quantum random walks.

Geoffrey Grimmett1, Svante Janson, Petra F Scudo

  • 1Statistical Laboratory, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK. g.r.grimmett@statslab.cam.ac.uk

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 5, 2004
PubMed
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Quantum random walks in multiple dimensions exhibit a weak limit theorem. Their position over time converges to an absolutely continuous distribution with bounded support, proven using Fourier transforms.

Area of Science:

  • Quantum mechanics
  • Probability theory
  • Mathematical physics

Background:

  • Quantum random walks (QRWs) are quantum analogues of classical random walks.
  • Previous studies primarily focused on one-dimensional QRWs, often using combinatorial or path integral methods.
  • A general framework for multi-dimensional QRWs and their limiting behavior was lacking.

Purpose of the Study:

  • To formulate and prove a general weak limit theorem for quantum random walks.
  • To extend the understanding of QRW behavior to higher dimensions.
  • To provide a rigorous mathematical foundation for the limiting distributions of QRWs.

Main Methods:

  • Development of a general weak limit theorem applicable to multi-dimensional QRWs.
  • Rigorous mathematical proof utilizing Fourier transform methods.

Related Experiment Videos

  • Analysis of the convergence of the position of the quantum random walk over time.

    • Main Results:

    • Demonstrated that the scaled position X(n)/n of a quantum random walk converges weakly as n approaches infinity.
    • Identified the limit as a distribution that is absolutely continuous and possesses bounded support.
    • The Fourier transform approach provided a simplified and generalized method compared to prior techniques.

    Conclusions:

    • The study establishes a fundamental limit theorem for quantum random walks in any dimension.
    • The findings reveal the nature of the limiting distribution, offering insights into the long-term behavior of these quantum systems.
    • The employed Fourier transform method offers a powerful and versatile tool for analyzing quantum random walks.