Topological Property Of Discrete-time Quantum Walk And Its Spread Property With Disorder

Compared to classical random walk,quantum walk as a quantum analogue of classical random walk,exhibits different features and advantages.Using the ballistic spreading of quantum walk,a universal computational primitive and the quantum algorithm can be designed.What's more,discrete-time quantum walk is a powerful tool for simulating topological phases of natural materials.In this thesis,we first give the concept of quantum walk and introduce the dynamics of two-period discrete-time quantum walk.Then,the topological properties of generalized split-step quantum walk are studied.Furthermore,considering two interacting particles and percolation in position space,the spreading properties are discussed for initial states with different symmetries.The model of split-step quantum walk is generalized and it can be equivalent to two successive operations of the standard discrete-time quantum walk under different coin parameters.The topological property of this generalized model is investigated.We demonstrate that the mean chiral displacement becomes proportional to the Zak phase in the infinite time limit,which means that the mean chiral displacement can be seen as dynamically topological invariant to describe the topological phase of split-step quantum walk.According to the mean chiral displacement,the topological phase diagram can be obtained.Furthermore,we consider the perturbation to the coin parameters and discuss its effect on the mean chiral displacement and topological bound states.It is shown that the mean chiral displacement oscillates with the time steps in both dynamical and static perturbation,and it tends to the static constant value as the increasing of time steps.This result is consistent with that of the ideal case,thus the mean chiral displacement is robust against perturbation.The peaked probability distribution at the boundary of two distinct topological phases still exist even consideringdisorder perturbation,so the Majorana state also is robust.The dynamical percolation can make single-particle quantum walk diffuse classically while the static percolation can make it localize at the neighbor of the initial site.We generalize to the case of two particle quantum walk.Considering interaction between two particles and percolation of the lattice simultaneously,their effects on spreading features are investigated.According to the symmetry for two bosons,fermions and classical indistinguishable particles,three kind of initial states are considered.We focus on the joint probability distributions,the average distance and the meeting probability to describe the global properties of quantum walk.The interaction between two particles leads to bosons anti-bunching and fermions bunching.The joint probability distributions of two particles spread more out over the walking region in the dynamical percolation,while they are concentrated on the neighbor of the initial position in the static case.For fermions and bosons initial states,both the curves of the meeting probability and that of the average distance occur crossing at a certain value of percolation probability,which is a result induced by the interplay between the interaction and percolation.