The Study Of Topological Property And Transmission For Non-unitary Discrete-time Quantum Walk

Quantum walk is the quantum counterpart of classical random walk.Compared with classical random walk,it has many advantages.For example,it can be used to design faster quantum algorithms due to its ballistic transmission.Discrete-time quantum walk becomes an effective tool to explore the topological properties of materials because it has very similar energy band structure to the topological insulators,and so on.In the discrete-time quantum walk,the polarization vector corresponding to the coin eigenstate can be obtained from the evolution operator.The winding number of the quantum walk,which describes its topological property,can be calculated by this polarization vector.In this paper,non-unitary quantum walk is introduced by the partial measurement operator,and its topological phase diagram labelled by the winding number in the two-dimensional parameter space is plotted.It is verified that there exist topologically protected edge states at boundaries between different topological phases.Moreover,the first dynamic moment or average displacement of quantum walk is calculated.Its variation with controllable parameters also can be used to describe topological phase transition.The transmission property of quantum walk is dependent on not only the evolution process determined by the evolution operator,but also the initial state.The initial state of discrete-time quantum walk is produced by a direct product of initial position state and initial coin state.For a certain initial coin state,it is found that the variation of the first dynamic moment with coin parameters can perfectly show the topological phase transition of system when the initial position state localizes at the origin.However,it is not the case for initial position state with Gaussian distribution.With the width of Gaussian distribution being larger,the description of topological transition by the first dynamic moment becomes degenerative.Furthermore,the average of first dynamicalmoment over all-possible coin initial states is performed,then the oscillation amplitude of the first dynamic moment reduces.Finally,we study the influence of coherence phase in coin initial state on the first dynamic moment.The magnitude of the first dynamic moment is changed but the jumping point is not changed.The description of topological transition is not affected regardless of the initial position state.