Stationary measure for two-state space-inhomogeneous quantum walk in one dimension

We consider the two-state space-inhomogeneous coined quantum walk (QW) in one dimension. For a general setting, we obtain the stationary measure of the QW by solving the eigenvalue problem. As a corollary, stationary measures of the multi-defect model and space-homogeneous QW are derived. The former is a generalization of the previous works on one-defect model and the latter is a generalization of the result given by Konno and Takei (2015).Comment: 15 pages, revised version, Yokohama Mathematical Journal (in press

Resolvent Methods for Quantum Walks with an Application to a Thue-Morse Quantum Walk

In this expository note, we discuss spatially inhomogeneous quantum walks in one dimension and describe a genre of mathematical methods that enables one to translate information about the time-independent eigenvalue equation for the unitary generator into dynamical estimates for the corresponding quantum walk. To illustrate the general methods, we show how to apply them to a 1D coined quantum walk whose coins are distributed according to an element of the Thue--Morse subshift.Comment: This paper is part of the proceedings volume for the Workshop on "Quantum Simulation and Quantum Walks" held in Yokohama, Japan in November of 201