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Entropy 2014, 16(3), 1501-1514; doi:10.3390/e16031501

Localization of Discrete Time Quantum Walks on the Glued Trees

1
Department of Information Systems Creation, Faculty of Engineering, Kanagawa University, Kanagawa, Yokohama 221-8686, Japan
2
Department of Applied Mathematics, Faculty of Engineering, Yokohama National University, Hodogaya, Yokohama 240-8501, Japan
3
Graduate School of Information Science, Tohoku University, Aoba, Sendai 980-8579, Japan
4
School of Physical Science and Technology, Soochow University, Suzhou 215006, China
5
Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea
*
Author to whom correspondence should be addressed.
Received: 4 December 2013 / Revised: 15 January 2014 / Accepted: 10 March 2014 / Published: 18 March 2014
(This article belongs to the Special Issue Advances in Information Theory)
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Abstract

In this paper, we consider the time averaged distribution of discrete time quantum walks on the glued trees. In order to analyze the walks on the glued trees, we consider a reduction to the walks on path graphs. Using a spectral analysis of the Jacobi matrices defined by the corresponding random walks on the path graphs, we have a spectral decomposition of the time evolution operator of the quantum walks. We find significant contributions of the eigenvalues, ±1, of the Jacobi matrices to the time averaged limit distribution of the quantum walks. As a consequence, we obtain the lower bounds of the time averaged distribution.
Keywords: discrete time quantum walks; Localization; glued tree; Jacobi matrix; spectral analysis; Orthogonal Polynomial; Chebyshev polynomial discrete time quantum walks; Localization; glued tree; Jacobi matrix; spectral analysis; Orthogonal Polynomial; Chebyshev polynomial
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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MDPI and ACS Style

Ide, Y.; Konno, N.; Segawa, E.; Xu, X.-P. Localization of Discrete Time Quantum Walks on the Glued Trees. Entropy 2014, 16, 1501-1514.

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