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

Abstract

In this paper we consider the one-dimensional quantum random walk $X_n^{\varphi }$ 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 $X_n^{\varphi }$. 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 $X_n^{\varphi }/n$ converges weakly to a limit $Z^{\varphi }$ as $n\to \mathrm{\infty }$, 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.