  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
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 X^{varphi}_n /n converges in distribution 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.
