In this paper, following the recent paper on Walk/Zeta Correspondence by the
first author and his coworkers, we compute the zeta function for the three- and
four-state quantum walk and correlated random walk, and the multi-state random
walk on the one-dimensional torus by using the Fourier analysis. We deal with
also the four-state quantum walk and correlated random walk on the
two-dimensional torus. In addition, we introduce a new class of models
determined by the generalized Grover matrix bridging the gap between the Grover
matrix and the positive-support of the Grover matrix. Finally, we give a
generalized version of the Konno-Sato theorem for the new class. As a
corollary, we calculate the zeta function for the generalized Grover matrix on
the d-dimensional torus.Comment: 23 pages. arXiv admin note: text overlap with arXiv:2104.1028