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

摘要

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