Mixing times of three-state quantum walks on cycles

Abstract

In this paper, we successfully obtain an explicit expression of the limit distribution [Formula: see text] of three-state quantum walks on cycles, the total variation distance between [Formula: see text] and the average probability [Formula: see text], and lower bound on the difference between two eigenvalues, among others. Based on the above conclusions, we finally get the mixing time [Formula: see text] of the quantum walk of Grover coin on the N-cycle. [Formula: see text] is the time required to characterize [Formula: see text] approaching [Formula: see text]. 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.

Quantum walk of two particles on the graphene structure graph with interaction

“…Quantum walk, also known as quantum random walk, it is a consequence of the quantization of classical random walks [1,2], which can be used as a powerful technological tool for constructing quantum algorithms [3][4][5] and simulating complex quantum systems. The existence of quantum walk gives it many properties that are superior to the classical ones, such as the familiar mixing [6,7] and hitting [8] times in the quantum context that are quadratic accelerations of the classical case. In addition, quantum walk have a wide range of applications in quantum computing [9], and quantum measurements [10].…”