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, 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 weakly 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.
Publié le : 2005-10-14
Classification:
quantum random walk,
the Hadamard walk,
limit theorems,
60F05,
60G50,
82B41,
81Q99
@article{1150287309,
author = {KONNO, Norio},
title = {A new type of limit theorems for the one-dimensional quantum random walk},
journal = {J. Math. Soc. Japan},
volume = {57},
number = {4},
year = {2005},
pages = { 1179-1195},
language = {en},
url = {http://dml.mathdoc.fr/item/1150287309}
}
KONNO, Norio. A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Japan, Tome 57 (2005) no. 4, pp. 1179-1195. http://gdmltest.u-ga.fr/item/1150287309/