Continuous-time quantum walk on integer lattices and homogeneous trees

TL;DR

This paper analyzes the behavior of continuous-time quantum walks on integer lattices and homogeneous trees, providing explicit asymptotic probabilities and distributions using generating function methods.

Contribution

It introduces a comprehensive analysis of quantum walks on various structures, including explicit asymptotic formulas for return probabilities and distributions.

Findings

Derived the limit of average probability distributions for isotropic walks on Z.

Computed asymptotic return probabilities at time t for all studied structures.

Provided generating function-based methods for analyzing quantum walks on complex graphs.

Abstract

This paper is concerned with the continuous-time quantum walk on Z, Z^d, and infinite homogeneous trees. By using the generating function method, we compute the limit of the average probability distribution for the general isotropic walk on Z, and for nearest-neighbor walks on Z^d and infinite homogeneous trees. In addition, we compute the asymptotic approximation for the probability of the return to zero at time t in all these cases.