Convergence of quantum random walks with decoherence

Journal Article · Sat Oct 15 04:00:00 EDT 2011 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.84.042317· OSTI ID:22080357

Shimao, Fan; Zhiyong, Feng; Yang, Wei-Shih ^[1]; Sheng, Xiong ^[2]

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1. Department of Mathematics Temple University, Philadelphia, Pennsylvania 19122 (United States)
2. Department of Mathematics and Sciences Edward Waters College, Jacksonville, Florida 32209 (United States)

In this paper, we study the discrete-time quantum random walks on a line subject to decoherence. The convergence of the rescaled position probability distribution p(x,t) depends mainly on the spectrum of the superoperator L{sub kk}. We show that if 1 is an eigenvalue of the superoperator with multiplicity one and there is no other eigenvalue whose modulus equals 1, then P(({nu}/{radical}(t)),t) converges to a convex combination of normal distributions. In terms of position space, the rescaled probability mass function p{sub t}(x,t){identical_to}p({radical}(t)x,t), x is an element of Z/{radical}(t), converges in distribution to a continuous convex combination of normal distributions. We give a necessary and sufficient condition for a U(2) decoherent quantum walk that satisfies the eigenvalue conditions. We also give a complete description of the behavior of quantum walks whose eigenvalues do not satisfy these assumptions. Specific examples such as the Hadamard walk and walks under real and complex rotations are illustrated. For the O(2) quantum random walks, an explicit formula is provided for the scaling limit of p(x,t) and their moments. We also obtain exact critical exponents for their moments at the critical point and show universality classes with respect to these critical exponents.

• OSTI ID: 22080357
• Journal Information: Physical Review. A, Journal Name: Physical Review. A Journal Issue: 4 Vol. 84; ISSN 1050-2947; ISSN PLRAAN
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CONVERGENCE
DISTRIBUTION
EIGENVALUES
GRAPH THEORY
PROBABILITY
QUANTUM DECOHERENCE
RANDOMNESS