15 research outputs found
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
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The stationary measure for diagonal quantum walk with one defect
This study is motivated by the previous work [14]. We treat 3 types of the
one-dimensional quantum walks (QWs), whose time evolutions are described by
diagonal unitary matrix, and diagonal unitary matrices with one defect. In this
paper, we call the QW defined by diagonal unitary matrices, "the diagonal QW",
and we consider the stationary distributions of generally 2-state diagonal QW
with one defect, 3-state space-homogeneous diagonal QW, and 3-state diagonal QW
with one defect. One of the purposes of our study is to characterize the QWs by
the stationary measure, which may lead to answer the basic and natural
question, "What the stationary measure is for one-dimensional QWs ?". In order
to analyze the stationary distribution, we focus on the corresponding
eigenvalue problems and the definition of the stationary measure.Comment: 10 page
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
Stationary Measures of Space-Inhomogeneous Three-State Quantum Walks on Line: Revisited
Of a quantum walk, its stationary measures play an important role in
understanding its evolution behavior. In this paper we investigate stationary
measures of two models of space-inhomogeneous three-state quantum walk on the
line. By using the method of reduced matrix, we find out stationary measures of
the two models under some mild conditions. Our results generalize the
corresponding ones existing in the literature