Mixing times of three-state quantum walks on cycles

In this paper, we successfully obtain an explicit expression of the limit distribution pi(nu) of three-state quantum walks on cycles, the total variation distance between pi(nu) and the average probability (p) over bar (nu)(T), and lower bound on the difference between two eigenvalues, among others. Based on the above conclusions, we finally get the mixing time T-epsilon of the quantum walk of Grover coin on the N-cycle. T-epsilon is the time required to characterize (p) over bar (nu)(T) approaching pi(nu). Our results show that the average probability of a three-state quantum walk on a cycle can approach its limit distribution faster than that of a two-state quantum walk, which might be of significance to quantum computation.