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As a first step of the study of the relation between the distribution of prime, reduced cycles of a graph and the limit theorem for the probability of a quantum walk on a graph, we present a trace formula with respect to the Grover matrix that is the transition matrix of the Grover walk on a graph. We define a zeta function with respect to the Grover matrix of a graph, and differentiate the logarithm of its zeta function of a regular graph. Furthermore, we obtain a trace formula with respect to the Grover matrix by a suitable complex integral of its logarithm differential.

The trace formula with respect to the Grover matrix of a graph