In this paper we consider the one-dimensional quantum random walk at time starting from initial qubit state determined by unitary matrix . We give a combinatorial expression for the characteristic function of . The expression clarifies the dependence of it on components of unitary matrix and initial qubit state . 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 converges weakly to a limit as , where has a density for . Moreover we discuss some known simulation results based on our limit theorems.
Norio KONNO. "A new type of limit theorems for the one-dimensional quantum random walk." J. Math. Soc. Japan 57 (4) 1179 - 1195, October, 2005. https://doi.org/10.2969/jmsj/1150287309