离散时间和连续时间量子随机行走的动力学研究

Discrete-time and continuous-time quantum (random) walks are two different theoretical models for quantum walks. Both of them have become hot topics in the scientific community due to their broad applications in the field of quantum computation、quantum information and condensed mater physics. In this project we will investigate the dynamical behavior of one-dimensional, two-dimensional and high-dimensional discrete-time and continuous-time quantum walks, including the probability distribution and its fluctuation, variance, mean square displacement and other basic statistical properties. We analyze the scaling behavior of return probability and calculate the precise value of Polya number, and try to determine the recurrence properties of quantum walks. Because the discrete-time quantum walk requires a "coin" to control the move in each step, we can study the impact of coin parameters on its relevant dynamics. In addition, we will compare the similarity and difference of the dynamical behavior between the two quantum walk models on various structures, reveal the influence of structure on the dynamics and shed some light on the possible relationship between the two theoretical models. Finally, we draw a comparison of the dynamical behavior bewteen quantum walk and its corresponding classical random walk, with the aim of obtaining some significant difference between coherent and noncoherent dynamics.

离散时间(Discrete-time)和连续时间(Continuous-time)量子随机行走(Quantum random walk)是两种不同的量子随机行走理论模型，在量子计算、量子信息和凝聚态物理中有广泛的应用，是近年来国际学术界研究的热点。本项目将研究一维，二维和高维规则结构上离散时间和连续时间量子随机行走的动力学行为，包括几率分布及其涨落，方差，方均位移等基本统计性质，分析回归几率的标度行为并计算Polya数，确定量子随机行走的回归性质。由于离散时间量子随机行走需要"硬币"来控制每步的移动，我们还将探讨"硬币"参数对其动力学性质的影响。比较不同结构上两种量子随机行走模型动力学性质的共同点和不同点，讨论结构对动力学的一些影响并揭示这两种模型之间可能的理论联系。最后将得到的量子随机行走动力学行为和它们对应的经典随机行走的动力学行为进行比较，得到相干和非相干动力学的一些显著区别。

本项目研究了离散时间量子随机行走(Discrete-time quantum walk)和连续时间量子随机行走(Continuous-time quantum walk)的动力学。采用定相积分近似方法和切比雪夫多项式方法推导了一维环，一维链及其变异结构上的概率分布及动力学变量。探讨"硬币"参数对离散时间量子随机行走动力学的影响，比较不同结构上两种量子随机行走模型动力学性质的共同点和不同点，讨论结构对动力学的一些影响并揭示这两种模型之间可能的理论联系。这些重要结构发表在国际和国内期刊上，完成了项目的预定目标和任务。