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Updated: Jul 30, 2026
07:46
Setting Limits on Supersymmetry Using Simplified Models
Published on: November 15, 2013
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
Summary
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.
- 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.
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