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In quantum computation theory, quantum random walks have been utilized bymany quantum search algorithms which provide improved performance over theirclassical counterparts. However, due to the importance of the quantumdecoherence phenomenon, decoherent quantum walks and their applications havebeen studied on a wide variety of structures. Recently, a unifiedtime-inhomogeneous coin-turning random walk with rescaled limitingdistributions, Bernoulli, uniform, arcsine and semi-circle laws as parametervaries have been obtained. In this paper we study the quantum analogue of thesemodels. We obtained a representation theorem for time-inhomogeneous quantumwalk on discrete infinite state space. Additionally, the convergence of thedistributions of the decoherent quantum walks are numerically estimated as anapplication of the representation theorem, and the convergence in distributionof the quantum analogues of Bernoulli, uniform, arcsine and semicircle laws arestatistically analyzed.

Time-inhomogeneous Quantum Walks with Decoherence on Discrete Infinite Spaces