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-n(phi) at time n starting from initial quoit state W determined by 2 x 2 unitary matrix U. We give a combinatorial expression for the characteristic function of X-n(phi). The expression clarifies the dependence of it on components of unitary matrix U and initial qubit state phi. 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(phi)/n converges weakly to a limit Z(phi) as n - infinity, where Z(phi) has a density 1/pi(1 - x(2))root 1 - 2x(2) for x is an element of (-1/root 2,1/root 2). Moreover we discuss some known simulation results based on our limit theorems.