Limit Probability Distributions For Time-inhomogeneous Quantum Walks

Quantum walks are quantum analogues of the classical random walk, which have found wide application in quantum algorithms, quantum information, quantum probability and biological physics due to their rapid diffusion speed. In recent years,time-homogeneous quantum walks have been intensively studied and many deep results have been obtained of them. In this dissertation, we mainly investigate the asymptotic behavior of a time-inhomogeneous two-state quantum walk and a timeinhomogeneous three-state quantum walk. By using the method of Fourier analysis,we obtain their limit probability distributions. The dissertation is organized as follows.In Chapter 1, we give a brief introduction to the background and current status for study of quantum walks. We also present some general concepts, facts and fundamental methods concerning quantum walks.In Chapter 2, we study a time-inhomogeneous two-state quantum walk recently introduced by Machida in a paper. We correct a mistake of Machida in his derivation and give the correct limit probability distributions of the walk. We also examine its localization properties.Finally in Chapter 3, we consider a time-inhomogeneous three-state quantum walk on the line. By using the method of Fourier analysis introduced by Grimmett et al, we obtain its limit probability distributions. We also offer some figures to help illustrate properties of its probability distributions.