  The quantum walk is a quantum counterpart of the classical random walk. On
the other hand, the absolute zeta function can be considered as a zeta function
over F_1. This paper presents a connection between the quantum walk and the
absolute zeta function. First we deal with a zeta function determined by a time
evolution matrix of the Grover walk on a graph. The Grover walk is a typical
model of the quantum walk. Then we prove that the zeta function given by the
quantum walk is an absolute automorphic form of weight depending on the number
of edges of the graph. Furthermore we consider an absolute zeta function for
the zeta function based on a quantum walk. As an example, we compute an
absolute zeta function for the cycle graph and show that it is expressed as the
multiple gamma function of order 2.
