Limit theorems and absorption problems for one-dimensional correlated random walks

There has recently been considerable interest in quantum walks in connection with quantum computing. The walk can be considered as a quantum version of the so-called correlated random walk. We clarify a strong structural similarity between both walks and study limit theorems and absorption problems for correlated random walks by our PQRS method, which was used in our analysis of quantum walks.Comment: 19 pages, revised versio

Quantum walks and elliptic integrals

Polya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional quantum walk.Comment: 8 pages, Journal-ref adde

The uniform measure for discrete-time quantum walks in one dimension

We obtain the uniform measure as a stationary measure of the one-dimensional discrete-time quantum walks by solving the corresponding eigenvalue problem. As an application, the uniform probability measure on a finite interval at a time can be given.Comment: 21 pages, revised version, Quantum Information Processing (in press

Quantum random walks in one dimension

This letter treats the quantum random walk on the line determined by a 2 times 2 unitary matrix U. A combinatorial expression for the mth moment of the quantum random walk is presented by using 4 matrices, P, Q, R and S given by U. The dependence of the mth moment on U and initial qubit state phi is clarified. A new type of limit theorems for the quantum walk is given. Furthermore necessary and sufficient conditions for symmetry of distribution for the quantum walk is presented. Our results show that the behavior of quantum random walk is striking different from that of the classical ramdom walk.Comment: 9 pages, journal reference adde

Localization of an inhomogeneous discrete-time quantum walk on the line

We investigate a space-inhomogeneous discrete-time quantum walk in one dimension. We show that the walk exhibits localization by a path counting method.Comment: 10 pages, 1 figure, minor corrections, Journal-ref added

A Path Integral Approach for Disordered Quantum Walks in One Dimension

The present letter gives a rigorous way from quantum to classical random walks by introducing an independent random fluctuation and then taking expectations based on a path integral approach.Comment: 9 pages, small corrections, journal reference adde

Continuous-time quantum walks on ultrametric spaces

We introduce a continuous-time quantum walk on an ultrametric space corresponding to the set of p-adic integers and compute its time-averaged probability distribution. It is shown that localization occurs for any location of the ultrametric space for the walk. This result presents a striking contrast to the classical random walk case. Moreover we clarify a difference between the ultrametric space and other graphs, such as cycle graph, line, hypercube and complete graph, for the localization of the quantum case. Our quantum walk may be useful for a quantum search algorithm on a tree-like hierarchical structure.Comment: 13 pages, small corrections, Journal-ref adde

Sojourn times of the Hadamard walk in one dimension

The Hadamard walk is a typical model of the discrete-time quantum walk. We investigate sojourn times of the Hadamard walk on a line by a path counting method.Comment: 14 pages, title changed, minor corrections, Quantum Information Processing (in press

Limit theorems and absorption problems for quantum random walks in one dimension

In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of one-dimensional quantum random walks determined by 2 times 2 unitary matrices using our PQRS method. The one type was introduced by Gudder in 1988, and the other type was studied intensively by Ambainis et al. in 2001. The difference between both types of quantum random walks is also clarified.Comment: 19 pages, small corrections, journal reference adde

A New Type of Limit Theorems for the One-Dimensional Quantum Random Walk

In this paper we consider the one-dimensional quantum random walk X^{varphi} _n at time n starting from initial qubit state varphi determined by 2 times 2 unitary matrix U. We give a combinatorial expression for the characteristic function of X^{varphi}_n. The expression clarifies the dependence of it on components of unitary matrix U and initial qubit state varphi. As a consequence of the above results, we present a new type of limit theorems for the quantum random walk. In contrast with the de Moivre-Laplace limit theorem, our symmetric case implies that X^{varphi}_n /n converges in distribution to a limit Z^{varphi} as n to infty where Z^{varphi} has a density 1 / pi (1-x^2) sqrt{1-2x^2} for x in (- 1/sqrt{2}, 1/sqrt{2}). Moreover we discuss some known simulation results based on our limit theorems.Comment: Final version; Journal-ref added; 14 pages; this arXiv version has no figure