Convergence of quantum random walks with decoherence
Abstract
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.
- Authors:
-
- Department of Mathematics Temple University, Philadelphia, Pennsylvania 19122 (United States)
- Department of Mathematics and Sciences Edward Waters College, Jacksonville, Florida 32209 (United States)
- Publication Date:
- OSTI Identifier:
- 22080357
- Resource Type:
- Journal Article
- Journal Name:
- Physical Review. A
- Additional Journal Information:
- Journal Volume: 84; Journal Issue: 4; Other Information: (c) 2011 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1050-2947
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONVERGENCE; DISTRIBUTION; EIGENVALUES; GRAPH THEORY; PROBABILITY; QUANTUM DECOHERENCE; RANDOMNESS
Citation Formats
Shimao, Fan, Zhiyong, Feng, Yang, Wei-Shih, and Sheng, Xiong. Convergence of quantum random walks with decoherence. United States: N. p., 2011.
Web. doi:10.1103/PHYSREVA.84.042317.
Shimao, Fan, Zhiyong, Feng, Yang, Wei-Shih, & Sheng, Xiong. Convergence of quantum random walks with decoherence. United States. https://doi.org/10.1103/PHYSREVA.84.042317
Shimao, Fan, Zhiyong, Feng, Yang, Wei-Shih, and Sheng, Xiong. 2011.
"Convergence of quantum random walks with decoherence". United States. https://doi.org/10.1103/PHYSREVA.84.042317.
@article{osti_22080357,
title = {Convergence of quantum random walks with decoherence},
author = {Shimao, Fan and Zhiyong, Feng and Yang, Wei-Shih and Sheng, Xiong},
abstractNote = {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.},
doi = {10.1103/PHYSREVA.84.042317},
url = {https://www.osti.gov/biblio/22080357},
journal = {Physical Review. A},
issn = {1050-2947},
number = 4,
volume = 84,
place = {United States},
year = {Sat Oct 15 00:00:00 EDT 2011},
month = {Sat Oct 15 00:00:00 EDT 2011}
}