中文摘要：

作为量子算法研究的一个基本工具,量子行走已经成为一个重要研究课题.在开放量子环境下,同质量子行走已经得到充分研究,包括其概率分布和中心极限定理.然而,对于高维格上且在异质环境下的开放量子行走的演化方程、概率分布和中心极限定理还未得到研究.在此基础上,该文提出高维格上异质开放量子行走以及它的演化方程,重点研究高维格上具有不同类型的量子运算的开放量子行走的概率分布和中心极限定理.首先给出高维格上量子系统的演化表达式,它不但适合于同质开放量子行走,也适合于异质开放量子行走.与已有的演化表达式相比,该结果更具有一般性.其次,利用傅里叶变换和逆变换给出开放量子行走的概率分布的计算公式,研究其对同质开放量子行走和异质开放量子行走的适用性.通过例子说明一维格上、二维格上异质开放量子行走的概率分布计算.最后,运用鞍差分序列中心极限定理,给出并证明高维格上异质开放量子行走的中心极限定理,说明一维格上同质开放量子行走的中心极限定理是它的一种特殊情况,并通过二维格上异质开放量子行走的实例给出求解极限分布的具体过程.

英文摘要：

As a basic tool for researching quantum algorithms, quantum walk has been an important aspect of the quantum computation. In an open quantum environment, homogenous quantum walk has all been studied, including its limit probability distribution and its central limit theorem. However, the evolution formula, the probability distribution and the central limit theorem have not been studied about open quantum walk with higher dimensional lattices and non-homogenous environment. Based on these works, we propose an open quantum walk on higher dimensional lattices and give its evolution formula, especially for one under non-homogenous environment, focus on its probability distribution and their central limit theorem with a different type of quantum operators. Firstly, we give an evolution formula of open quantum walk on higher dimensional lattices, show that it not only adapts to homogenous quantum walk~ but also adapts to non-homogenous quantum walk. Compared with the existing evolution formulas, this result is more general. Secondly, by using the fourier transform and the fourier inverse transform, we give a computational formula for the probability distribution of open quantum walk, study its applicability for homogenous quantum walk and non-homogenous quantum walk. To present by examples, we show a computational process about their probability distributions of non-homogenous quantum walks on 1-dimensional lattice and 2-dimensional lattice. Finally, according to the central theorem for maringale difference sequence, we prove a central limit theorem for non-homogenous open quantum walk on higher dimensional lattices under certain condition, show that a central limit theorem is its special case about a one-dimensional lattice homogenous open quantum walk, illustrate a specific process of calculating the limit distribution.