A zeta function related to the transition matrix of the discrete-time quantum walk on a graph

Abstract

We present the structure theorem for the positive support of the cube of the Grover transition matrix of the discrete-time quantum walk (the Grover walk) on a general graph G under some condition. Thus, we introduce a zeta function on the positive support of the cube of the Grover transition matrix of G, and present its Euler product and its determinant expression. As a corollary, we give the characteristic polynomial for the positive support of the cube of the Grover transition matrix of a regular graph, and so obtain its spectra. Finally, we present the poles and the radius of the convergence of this zeta function.