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

In this paper we consider the one-dimensional quantum random walk Xφ n at time n starting from initial qubit state φ determined by 2× 2 unitary matrix U . We give a combinatorial expression for the characteristic function of Xφ n . The expression clarifies the dependence of it on components of unitary matrix U and initial qubit state φ. 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 Xφ n /n converges in distribution to a limit Zφ as n → ∞ where Zφ has a density 1/π(1−x2) √ 1− 2x2 for x ∈ (− √ 2/2, √ 2/2). Moreover we discuss some known simulation results based on our limit theorems.