Localization of Discrete Time Quantum Walks on the Glued Trees

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