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, 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.