Localization of Discrete Time Quantum Walks on the Glued Trees

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.

[1]  Philippe Biane,et al.  Quantum Potential Theory , 2008 .

[2]  Andris Ambainis,et al.  One-dimensional quantum walks , 2001, STOC '01.

[3]  M. Szegedy,et al.  Quantum Walk Based Search Algorithms , 2008, TAMC.

[4]  Daniel A. Spielman,et al.  Exponential algorithmic speedup by a quantum walk , 2002, STOC '03.

[5]  J. B. Wang,et al.  A classical approach to the graph isomorphism problem using quantum walks , 2007, 0705.2531.

[6]  Etsuo Segawa,et al.  Localization of quantum walks induced by recurrence properties of random walks , 2011, 1112.4982.

[7]  Aharonov,et al.  Quantum Walks , 2012, 1207.7283.

[8]  K. Birgitta Whaley,et al.  Quantum random-walk search algorithm , 2002, quant-ph/0210064.

[9]  S. D. Berry,et al.  Two-particle quantum walks: Entanglement and graph isomorphism testing , 2011 .

[10]  Jingbo B. Wang,et al.  Physical Implementation of Quantum Walks , 2013 .

[11]  Julia Kempe,et al.  Quantum random walks: An introductory overview , 2003, quant-ph/0303081.

[12]  Vivien M. Kendon,et al.  Decoherence in quantum walks – a review , 2006, Mathematical Structures in Computer Science.

[13]  Andris Ambainis,et al.  Quantum walk algorithm for element distinctness , 2003, 45th Annual IEEE Symposium on Foundations of Computer Science.

[14]  Salvador Elías Venegas-Andraca,et al.  Quantum walks: a comprehensive review , 2012, Quantum Information Processing.

[15]  Timothy C. Ralph,et al.  Entanglement dynamics and quasi-periodicity in discrete quantum walks , 2011, 1102.4385.

[16]  Chris D. Godsil,et al.  Average mixing of continuous quantum walks , 2011, J. Comb. Theory, Ser. A.

[17]  Andris Ambainis,et al.  Quantum walks on graphs , 2000, STOC '01.

[18]  Etsuo Segawa,et al.  Localization of the Grover Walks on Spidernets and Free Meixner Laws , 2012, 1206.4422.

[19]  A. Hora,et al.  Quantum Probability and Spectral Analysis of Graphs , 2007 .

[20]  S. D. Berry,et al.  Quantum-walk-based search and centrality , 2010, 1010.0764.

[21]  Eric Bach,et al.  Noninteracting multiparticle quantum random walks applied to the graph isomorphism problem for strongly regular graphs , 2012, 1206.2999.

[22]  Albert H. Werner,et al.  Asymptotic evolution of quantum walks with random coin , 2010, 1009.2019.

[23]  Etsuo Segawa,et al.  Time averaged distribution of a discrete-time quantum walk on the path , 2011, Quantum Inf. Process..