Quantum Walks On Several Types Of Graphs From A View Point Of Quantum Probability

Quantum walk is the product of quantization of classical random walk.Its construction provides theoretical and technical support for quantum algorithms.It has profound theoretical significance and broad application prospects in the development of quantum computing.In the context of quantum computing,we are interested in quantum walk on the graph from the perspective of quantum probabilityThis thesis discusses quantum walk on several types of graphs based on quantum probability,i.e.the quantum decomposition of the adjacency matrix A of graph,regard adjacency matrix A as Hamiltonian which is a real symmetric matrix with elements 0 or 1,so we regard e-itA as an unbiased evolution operator,which is related to the calculation of probability amplitude.Firstly,we introduce the research background of quantum walk and the development of quantum probability.Secondly,the concepts related to graph theory are given and the problems of graph adjacency matrix,spectral distribution,quantum decomposition of random variables,asymptotic spectral distribution of growth graph,and intersection array of distance-regular graph are discussed.This provides a quantum probability basis for the subsequent study of quantum walk on graphs.Finally,we study quantum walk on some finite and infinite graphs,such as complete bipartite graphs,binary tree and so on.We analyze the walk problem on hypercube with the help of Cartesian product structure,and consider the probability amplitude on the star graph with the directed bond.