The Dynamic Of Discrete-Time Quantum Walks

Shor's fast factoring algorithm and Grover search algorithm have widely drawn people's attention to the studies of quantum computation and quantum information.These two kinds of algorithms suggest that quantum computation has unparallelled computational power when compared with its classical counterpart.In recent years,people have show emphasis on the studies of quantum walks,which is the quantum counterpart of classical random walks.Just as the extensive application of classical random walks on classical algorithms,quantum walks has widely been used in constructing efficient quantum algorithms.In this paper,we firstly introduce the theoretical and experimental development of quantum walks,and the numerical and analytical method to cope up with it.Then we discuss different features between quantum walks on disorder medium and the ordinary quantum walks,i.e.constant coin operation.Through numerical experiment,we find that the spreading speed of quantum walks on disorder medium is much lower than that in standard quantum walks,even lower than that in classical random walks.It presents”localized” feature.Then we introduce the Anderson Localization which mostly represents the ”localized” feature.Under the condition of static disorder,the quantum walker will be bounded within a small neighbouring of the starting point and the upper bound of the deviation has no concern with the number of steps quantum walker takes.Considering that the real-world network is finite and bounded we analyze the localization of quantum walks on finite graph.We first vectorize the probability distribution of a quantum walker in each nodes.Then we compute out the probability distribution vectors of quantum walks in infinite and finite graphs in the presence of static disorder respectively,and get the distance between these two vectors.We find that when the steps taken are small and the boundary condition is tight,the localization between the infinite and finite cases is of great differences.However,the difference is negligible when the steps taken are large or the boundary condition is loose.It means quantum walks on one-dimensional finite graph may also suffer from localization in the presence of static disorder.Our approach and results can be generalized to analyse the localization of quantum walks in higher-dimensional cases.This paper is helpful in understanding the dynamic of quantum walks and its analytical method.Also it makes you familiar with the features quantum walks shows under different conditions.The numerical study of localization on finite graphs is instructive to its correspongding experimental simulation.