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.
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https://hal.archives-ouvertes.fr/hal-00954344
Contributor : Alain Joye <>
Submitted on : Saturday, March 1, 2014 - 3:15:50 PM
Last modification on : Thursday, January 11, 2018 - 6:12:14 AM

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  • HAL Id : hal-00954344, version 1
  • ARXIV : 1307.2772

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Alain Joye, Laurent Marin. Spectral Properties of Quantum Walks on Rooted Binary Trees. Journal of Statistical Physics, Springer Verlag, 2014, 155, pp.415-439. ⟨hal-00954344⟩

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