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arXiv · 1911.06060

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

Norio Konno·Iwao Sato·Etsuo Segawa
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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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Norio Konno, Iwao Sato, Etsuo Segawa. 2019-11-14. A zeta function related to the transition matrix of the discrete-time quantum walk on a graph. https://arxiv.org/abs/1911.06060

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