  In quantum computation theory, quantum random walks have been utilized by
many quantum search algorithms which provide improved performance over their
classical counterparts. However, due to the importance of the quantum
decoherence phenomenon, decoherent quantum walks and their applications have
been studied on a wide variety of structures. Recently, a unified
time-inhomogeneous coin-turning random walk with rescaled limiting
distributions, Bernoulli, uniform, arcsine and semi-circle laws as parameter
varies have been obtained. In this paper we study the quantum analogue of these
models. We obtained a representation theorem for time-inhomogeneous quantum
walk on discrete infinite state space. Additionally, the convergence of the
distributions of the decoherent quantum walks are numerically estimated as an
application of the representation theorem, and the convergence in distribution
of the quantum analogues of Bernoulli, uniform, arcsine and semicircle laws are
statistically analyzed.
