Quantum Random Walk On Several Types Of Graphs

As the quantum counterpart of classical random walk,quantum random walk has become a powerful tool for people to study quantum computation in recent years.This paper mainly uses the model of quantum random walk and discusses the problem of quantum random walk on several types of graphs.In chapter 1,we focus on the background and research status of quantum random walk,and give some basic mathematical concepts related to it.In Chapter 2,we take the two-state quantum random walk on line as a model and construct a new shift operator.By standard unitary evolution and the principle of Fourier transform,we further discuss the one-dimensional three-state quantum random walk,the analytical expression of the state of the walker at time t is obtained.In Chapter 3,we use a shift operator without the flip-flop action,and the analytical expression of the state of the walker at time t is obtained,this result is consistent with the shift operator with the flip-flop action.In Chapter 4,we employ the principle of Fourier transform and introduce a set of invariant basis,some relevant properties of quantum random walk on a hypercube are obtained.In Chapter 5,we introduce the quantum random walk on the N-ary tree,the state space has no need for coin spaces,it imposes no constraints on the evolution operator other than unitarity,we further use z transform,regeneration summation and path enumeration to discuss the problems of quantum random walk on the N-ary tree.