Dynamics And Topological Properties Of The Periodic Discrete-time Time-dependent Quantum Walk

Quantum walk is a generalization of classical random walk in quantum mechanics.It has a faster propagation speed than classical random walk under the influence of quantum interference effect,and is widely used in quantum computing,quantum simulation and quantum information transmission.In recent years,there has been a great deal of interest in the use of discrete-time quantum walk simulation and the exploration of the topological properties of physical systems.In this paper,the basic concept and classification of quantum walk are first given,and topological invariants such as topological winding number are introduced.Then,the transmission characteristics of discrete-time quantum walk under the action of phase operators are studied,and the topological properties of three-period quantum walk under the action of two phase operators and the dynamic evolution process under the action of time-dependent phase operators are discussed.The phase operator is introduced to study the discrete-time quantum walk under the two conditions,respectively.It is found that the probability distribution and transmission characteristics are the same as the standard discrete-time quantum walk under the action of the phase operator without time.However,when the phase operator is constant,the probability distribution shows oscillatory behavior.According to the variance calculated within 100 steps of evolution,it is found that the quantum walk with time phase action deviates from the initial position to a higher degree than the quantum walk without time phase action.In particular,by introducing two time-dependent phase operators into the discrete-time quantum walk,the probability distribution and variance of different distribution characteristics can be obtained by adjusting the parameters of time-dependent phase,which is of great significance for quantum information transmission.The dynamic evolution and topological properties of the system are discussed by inserting phase operators into each flat step of the periodic quantum walk.The three-period quantum walk model is taken as an example,and the results are extended to more periodic cases.Firstly,the band structure of the three-period quantum walk system is calculated,and the relationship between the topological winding number and the phases accumulated during the walking is obtained.Then the phase is changed into a function of time,and the dynamics of the three-period discrete-time quantum walk under the action of the time-dependent phase operator dependent on evolution steps are studied.It is found that the probability distribution of quantum walk is represented by Bloch oscillation,especially the number of turning points of oscillation is equal to the winding number of system,and the topological properties of the system are characterized from the perspective of system dynamics.Finally,we discuss that the same result can be obtained by four-period quantum walk,which can be extended to quantum walk with more than one period.