Discrete quantum walk model and its applications.

In this thesis, discrete quantum walk model is examined and presented some applications of this model. Quantum walk is a mathematical model which is used for designing quantum algorithms and explaining quantum diffusion processes. In order to use the quantum walk's potential fully and compose efficient quantum algorithms, it is important to know its properties and understand its behavior against to decoherence problem. A lot of studies of the quantum walk's dynamics in the absence and presence ofdecoherence have been made and various decoherence models have been presented. Studies of this subject have still continued. Therefore, in this thesis, quantum walk and decoherence problems have been examined. In the first study, quantum Hadamard, Fourier and Grover walks have been analyzed in two dimensional trapped lattice and found that generated decoherence in Hadamard walk is more smaller thangenerated in Fourier and Grover walks. In the second study, using the same but three dimensional trapped lattice model with Hadamard walk, the dependence of dimension of decoherence has beeninvestigated and relations between decoherence and lattice dimension have been found. In the last study, survival probability of quantum walkers has been examined in trapped lattice and found thatevaluation of the time dependence of the survival probability of quantum walkers obeys the stretched exponential law.