Spectral Properties of Quantum Walks on Rooted Binary Trees

TL;DR

This paper investigates the spectral characteristics of quantum walks on rooted binary trees, revealing conditions under which the spectrum is purely absolutely continuous or has other spectral types, based on the coin matrix properties.

Contribution

It provides a detailed spectral analysis of quantum walks on rooted binary trees, especially characterizing the spectrum for circulant and orthogonal circulant coin matrices.

Findings

Circulant unitary coin matrices lead to spectra with no singular continuous component.

Orthogonal circulant matrices generally produce absolutely continuous spectra.

Four specific coin matrices result in pure point spectra.

Abstract

We define coined Quantum Walks on the infinite rooted binary tree given by unitary operators $U(C)$ on an associated infinite dimensional Hilbert space, depending on a unitary coin matrix $C\in U(3)$, and study their spectral properties. For circulant unitary coin matrices $C$, we derive an equation for the Carath\'eodory function associated to the spectral measure of a cyclic vector for $U(C)$. This allows us to show that for all circulant unitary coin matrices, the spectrum of the Quantum Walk has no singular continuous component. Furthermore, for coin matrices $C$ which are orthogonal circulant matrices, we show that the spectrum of the Quantum Walk is absolutely continuous, except for four coin matrices for which the spectrum of $U(C)$ is pure point.