Continuous-Time Quantum Walk on Integer Lattices and Homogeneous Trees

This paper is concerned with the continuous-time quantum walk on ℤ, ℤd, 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 ℤd 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.