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

2019/11/14 by Norio Konno, Konno, Norio, Iwao Sato +3
Computer Science · #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum-Dot Cellular Automata

paper · pdf · doi:10.48550/arxiv.1911.06060

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 same 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.

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