Spectral Properties of Quantum Walks on Rooted Binary Trees

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 Carathéodory 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.

References in corpus (4)

• Quantum Walk in Position Space with Single Optically Trapped Atoms
• Three-photon bosonic coalescence in an integrated tritter
• Open Quantum Random Walks
• Asymptotic evolution of quantum walks with random coin

Cited by in corpus (4)

• Spectral Stability of Unitary Network Models
• Quantum walks induced by Dirichlet random walks on infinite trees
• Spectral and scattering properties of quantum walks on homogenous trees of odd degree
• Dynamical Localization and Transport properties of Quantum Walks on the hexagonal lattice