Localization of Discrete Time Quantum Walks on the Glued Trees

Localization of discrete-time quantum walks on a half line via the CGMV method

We study discrete-time quantum walks on a half line by means of spectral analysis. Cantero et al. [1] showed that the CMV matrix, which gives a recurrence relation for the orthogonal Laurent polynomials on the unit circle [2], expresses the dynamics of the quantum walk. Using the CGMV method introduced by them, the name is taken from their initials, we obtain the spectral measure for the quantu...

Glued trees algorithm under phase damping

We study the behaviour of the glued trees algorithm described by Childs et al. in [STOC ‘03, Proc. 35 ACM Symposium on Theory of Computing (2004) 59] under decoherence. We consider a discrete time reformulation of the continuous time quantum walk protocol and apply a phase damping channel to the coin state, investigating the effect of such a mechanism on the probability of the walker appearing ...

Limit theorems for discrete-time quantum walks on trees

We consider a discrete-time quantum walk Wt given by the Grover transformation on the Cayley tree. We reduce Wt to a quantum walk Xt on a half line with a wall at the origin. This paper presents two types of limit theorems for Xt. The first one is Xt as t → ∞, which corresponds to a localization in the case of an initial qubit state. The second one is Xt/t as t → ∞, whose limit density is given...

Limit theorems for the discrete-time quantum walk on a graph with joined half lines

We consider a discrete-time quantum walk Wt,κ at time t on a graph with joined half lines Jκ, which is composed of κ half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and sho...

LOCALIZATION OF QUANTUM WALKS ON TREES WITH DISORDER by STEVEN JACKSON