任意初始态下离散量子随机行走的平均位置

Average Position of Discrete Time Quantum Walks with an Arbitrary Initial State

用模拟及傅里叶变换等数学方法,研究离散量子随机行走的平均值与初始态和硬币算符之间的关系。研究发现:在固定θ=β=π/4时,离散量子随机行走平均位置和其他参数满足余弦函数;而且如果选择参数φ-α-γ=π/2时,平均位置和初始态θ的关系也满足余弦函数。再通过傅里叶变换反向证明,得到任意初始态和硬币算符下,离散量子随机行走平均位置与初始态和硬币算符的数学表达式。只要已知初始态为0L〉和(0L〉+0R〉)2/2,硬币算符U(0,β,0)下的平均位置,就可以按照公式计算得到任意初始态和U(2)算符下的离散量子随机行走的平均位置。

The relationship between the average position of the discrete time quantum walk and the initial state and the coin operator is studied by means of computer simulation and Fourier transform. It is found that the average position and other parameters of the discrete time quantum walk can be used to satisfy the cosine function while the parameters θ=β=π/4 or φ - α -γ =π/2 . Then , by means of mathematical methods suchas Fourier transform , the mathematical expression of the average position of discrete time quantum walk and the initial state and the coin operator is obtained under any initial state and coin operator. Finally , as long as the average position of discrete time quantum walk is known under the initial state ｜0L〉和 （｜0L〉＋ 0R〉） V2/2 with coin operator U（ 0,β, 0） , we can calculate the average position of quantum random walk under any initialstate and operator.