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 $\kappa ,{\kappa }^{\prime }$ 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.

29 pages, 1 figure