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arXiv:1312.1149 · Entropy

Localization of Discrete Time Quantum Walks on the Glued Trees

Yusuke Ide, Norio Konno, Etsuo Segawa, Xin-Ping Xu

Published Mar 18, 2014• 9 citations
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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.

DOI
10.3390/e16031501

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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.
In this paper, we consider the time averaged distribution of discrete time quantum walks on the glued trees. In order to analyse 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 spectral decomposition of the time evolution operator of the quantum walks. We find significant contributions of the eigenvalues ±1\pm 1±1 of the Jacobi matrices to the time averaged limit distribution of the quantum walks. As a consequence we obtain lower bounds of the time averaged distribution.
In this paper, we consider a spectral analysis of the Correlated Random Walk (CRW) on the path. We apply an analytical method for the Quantum Walk to CRW. For the isospectral coin cases, we obtain all of the eigenvalues and the corresponding eigenvectors of the time evolution operator of CRW. We show that most of them are described by the eigenvalues and the eigenvectors of the Jacobi matrix related to the random walks on the path (the birth and death chains). The remaining eigenvector corresponding to an eigenvalue 1, is characterized by the stationary measure of the same birth and death chain. As a consequence, we derive the limiting distribution.
In this paper, we analytically investigated the transmission properties of continuous time quantum walks through two types of glued-tree graphs. By mapping the glued-tree graphs onto a non-uniform lattice line, we obtained their bound eigen-energies, which provided a clear and quantitative understanding of the associated resonant transmission behaviours. We also observed interesting zero-mode resonant transmissions, which correspond to zero-energy bound states, for both types of glued-tree graphs with classical Markov transition matrix parameters. By changing the system parameters at the glued nodes, we are able to control the position of the resonant tunnelling peaks.
There is a property called localization, which is essential for applications of quantum walks. From a mathematical point of view, the occurrence of localization is known to be equivalent to the existence of eigenvalues of the time evolution operators, which are defined by coin matrices. A previous study proposed an approach to the eigenvalue problem for space-inhomogeneous models using transfer matrices. However, the approach was restricted to models whose coin matrices are the same in positions sufficiently far to the left and right, respectively. This study shows that the method can be applied to extended models with periodically arranged coin matrices. Moreover, we investigate localization by performing the eigenvalue analysis and deriving their time-averaged limit distribution.
Quantum walks, both discrete (coined) and continuous time, form the basis of several recent quantum algorithms. Here we use numerical simulations to study the properties of discrete, coined quantum walks. We investigate the variation in the entanglement between the coin and the position of the particle by calculating the entropy of the reduced density matrix of the coin. We consider both dynamical evolution and asymptotic limits for coins of dimensions from two to eight on regular graphs. For low coin dimensions, quantum walks which spread faster (as measured by the mean square deviation of their distribution from uniform) also exhibit faster convergence towards the asymptotic value of the entanglement between the coin and particle's position. For high-dimensional coins, the DFT coin operator is more efficient at spreading than the Grover coin. We study the entanglement of the coin on regular finite graphs such as cycles, and also show that on complete bipartite graphs, a quantum walk with a Grover coin is always periodic with period four. We generalize the 'glued trees' graph used by Childs et al (2003 Proc. STOC , pp 59–68) to higher branching rate (fan out) and verify that the scaling with branching rate and with tree depth is polynomial.