Quantum walks driven by quantum coins with two multiple eigenvalues

We consider a spectral analysis on the quantum walks on graph G = (V, E) with the local coin operators {C-u}(u is an element of v) and the flip flop shift. The quantum coin operators have commonly two distinct eigenvalues kappa, kappa' and p = dim(ker(kappa - C-u)) for any u is an element of V with 1 <= p <= delta(G), where delta(G) is the minimum degrees of G. We show that this quantum walk can be decomposed into a cellular automaton on l(2)( V; C-p) whose time evolution is described by a self adjoint operator T and its remainder. We obtain how the eigenvalues and its eigenspace of T are lifted up to as those of the original quantum walk. As an application, we express the eigenpolynomial of the Grover walk on Zd with the moving shift in the Fourier space.