Quantum walks driven by quantum coins with two multiple eigenvalues

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