The Quantum Walk Of Particle On Lattice Cycle

Quantum walk is the quantum version of classical random walk. It is not only important to designing the quantum algorithms but also very useful for modeling the dynamic evolution of complex system. In this thesis the definations of continuous-time quantum walk and discrete-time quantum walk are present firstly. Their differences with classical random walk and some properties such as probability distribution, the standard deviation of the position, the probability of return to the initial position, the average mixing time and the Fourier transform method on an infinite line are introduced.Next we study the scattering dynamics of time discrete quantum walk on the closed cycle graph. The scattering potential is substituted by the parameter in coin operator. The effects of single potential and the double potential on the probability distribution are discussed. When the number of nodes on the cycle is eight, the periodic evolution of probability at each node on uniform cycle is destroyed by single potential, but it is recovered under the effect of double potential and the period is enlarged twice. For the case of two non-interaction particle on the cycle with two potential barriers, the evolution of their joined probability is also periodic regardless of their initial state being entangled or non-entangled. These characteristics of quantum walk are dependent of the number of node in cycle, for 6-node cycle the probability distribution has no periodic evolution.Finally, we use the measurement-induced disturbance to characterize the quantum correlation between position states of two particles. It is shown that if two particles have no interaction and their position initial states are uncorrelated but their coin states are entangled, the quantum correlation can transfer from coin states to position states with time evolution. The periodic evolution of quantum correlation between position states of two particles on uniform cycle can be destroyed by single node potential, but it is recovered under the effect of double potential. The quantum correlations of position states of two interacting particles on an uniform cycle can be produced whether the initial coin state is entangled or not. The interaction between two particles make the quantum correlations appear rapidly at the beginning. But the interaction does not lead to the position of the particle correlations with the initial position of the two states showing period evolution, there will be zero correlation appears. When the consideration of environmental noise, discover whether the two particle have interaction, interference pattern associated with the same location, but the amplitude will continue to decay with time, the greater the noise level, the greater reduce of amplitude,then decreased to zero asymptotically.