Spreading Property And Absorption Probability Of Discrete-time Quantum Walk

Quantum walk is quantum counterpart of classical random walk.It is not only a general model for building quantum algorithms and realizing universal quantum computation,but also an operational platform to simulate the dynamical characters of quantum system.So quantum walk can be applied in various fields of natural science and has been studied widely.In this paper,some basic concepts of classical random walks and quantum walks are first introduced and stationary measurement of discrete-time quantum walk is given.Then,single-point position defects and absorbing boundaries are included in one-dimensional infinite line system,to investigate the effects of single-point position defects on the spreading properties of discrete-time quantum walk and the effects of absorbing boundaries on the absorption probability of discrete-time quantum walk.For one-dimensional structure of infinite lattice,a single point defect is introduced by using a special coin operator on a certain position site.A parameter meaning reflecting amplitude is included in the special coin operator,which breaks the translational symmetry and results in localized state appearing at the neighborhood of defect site.The eigen bound solution of discrete-time quantum walk is obtained by solving the eigen equation of two-step unitary evolution operator.Also we numerically calculate the probability distribution and position deviation to study the spreading properties of quantum walk on one-dimension lattice line with single-point defect.The results show that the localization occurs only in some ranges of reflecting parameter and the strength of localization is determined by the projection of initial state on eigen bound states.Replacing the single-point position defect by a absorbing wall,which can be regarded as a semi-infinite lattice with reflection at the surface and unidirectional transmissions inside,we assume to put two absorbing walls on one-dimensional lattice and the initial site of quantum walk in between.The influences of reflecting parameter in coin operator,the location of the absorbing wall and the initial state on the absorption probability of absorbing boundary to the quantum walk are investigated.It is shown that if left absorbing boundary and right absorbing boundary are stationary,the absorption probability of left absorbing boundary decreases with the reflection parameter,and in the long time limit this probability tends to a asymptotic value.When left absorbing boundary is stationary but right absorbing boundary moves,compared to the results of the case with one stationary boundary,the moving effects of right boundary increases absorption probability of left boundary in some values of the reflection parameter,while the absorption probability of left boundary decreases in other values.