Quantum walks driven by quantum coins with two multiple eigenvalues

Abstract

We consider a spectral analysis on the quantum walks on graph $$G=(V,E)$$ with the local coin operators $$\{C_u\}_{u\in V}$$ 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 $$u\in V$$ with $$1\le p\le \delta (G)$$ , where $$\delta (G)$$ is the minimum degrees of G. We show that this quantum walk can be decomposed into a cellular automaton on $$\ell ^2(V;\mathbb {C}^p)$$ 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 $$\mathbb {Z}^d$$ with the moving shift in the Fourier space.