Quantum walks driven by quantum coins with two multiple eigenvalues

TL;DR

This paper analyzes quantum walks on graphs with specific coin operators having two eigenvalues, decomposing them into cellular automata and exploring their spectral properties, including applications to the Grover walk.

Contribution

It introduces a spectral decomposition method for quantum walks driven by coins with two eigenvalues, linking their spectra to cellular automata and applying it to the Grover walk.

Findings

Quantum walks can be decomposed into cellular automata with a self-adjoint operator.

Eigenvalues and eigenspaces of the decomposed operator relate to the original walk.

Eigenpolynomial of the Grover walk on  lattice expressed in Fourier space.

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 $1\leq p\leq \delta(G)$, 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.