## Journal of the Mathematical Society of Japan

### A new type of limit theorems for the one-dimensional quantum random walk

Norio KONNO

#### Abstract

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.

#### Article information

Source
J. Math. Soc. Japan, Volume 57, Number 4 (2005), 1179-1195.

Dates
First available in Project Euclid: 14 June 2006

https://projecteuclid.org/euclid.jmsj/1150287309

Digital Object Identifier
doi:10.2969/jmsj/1150287309

Mathematical Reviews number (MathSciNet)
MR2183589

Zentralblatt MATH identifier
1173.81318

#### Citation

KONNO, Norio. A new type of limit theorems for the one-dimensional quantum random walk. J. Math. Soc. Japan 57 (2005), no. 4, 1179--1195. doi:10.2969/jmsj/1150287309. https://projecteuclid.org/euclid.jmsj/1150287309

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