论文总字数：23447字

摘 要

近些年来，随着科学技术的进步，人类社会进入信息化时代，不管是科学研究还是日常生活，人们对计算的要求也越来越高，都希望在经典算法的基础上有所提升，研究出更为高效快捷且安全系数更高的量子算法。量子算法有着比经典计算机更为快速高效的能力。基于此，设计一个高效的，且由量子计算特点决定的量子算法成为了一个重要研究方向。

我们将量子行走描述为：一个具有两个自由度的粒子，由演化算符决定行走方向，在它行走过程中会发生波函数的演化。演化矩阵可以看成是由两个矩阵组成：作用于自旋空间的旋转矩阵C和通过自旋之后状态改变粒子行走方向的S矩阵，自旋矩阵的演化只由一个参数决定，当我们保持这个参数不变的时候，行走表现为均匀量子行走。有势垒的量子行走与均匀量子行走的区别在于其条件移动算符S随时间或空间发生变化，决定S算符的两个参量分别为α和β。，其中α为粒子成功隧穿并使其自旋态发生变化的幅度，β为保持不变的幅度，即作用完一步后，步行者的状态和自旋态不变。有势垒的量子行走其演化算符可以表示为。有均匀势垒的量子行走的一个完整步骤是粒子首先在C算符的作用下对其自旋态进行一个旋转，其次在S算符作用下进行移动。其中α，β不随时间发生变化。具有非均匀势垒的量子行走则α和β随时间发生变化。

本文主要通过数值计算的方法研究了通过两类广义Fibonacci序列和两种无序序列生成S算符，来产生具有非均匀势垒的量子行走的波包扩散行为，主要研究了S算符中两个参数αβ随时间变化的行走。通过计算导出波包扩散的概率分布图像，对标准方差拟合幂指数，并进行比较对照。从第一个物理量粒子波包扩散概率分布p来看，我们发现在第一类序列下，粒子波包扩散概率分布情况与经典随机行走类似；第二类序列下，粒子波包扩散概率分布情况则与均匀量子行走类似。二元无序情况下粒子波包扩散概率分布情况也与均匀量子行走类似；多元无序情况下则与经典随机行走相似。从另一个物理量标准方差随时间变化来看，序列下，波包扩散情况为亚弹道扩散；下，波包扩散情况为弹道扩散。二元无序情况下粒子波包扩散接近于弹道扩散；而多元无序情况下与经典随机行走接近，趋向正常扩散。

关键词：量子行走 非均匀势垒 无序势垒 亚弹道扩散 波包扩散

Abstract

In recent years, with the advancement of science and technology, human society has entered the information age. Whether it is scientific research or daily life, people have higher and higher requirements for computing. They all hope to improve on the basis of classic algorithms. To produce more efficient and faster quantum algorithms with higher safety factor. Because of this, the application of quantum walking in quantum algorithms and experiments has also developed more rapidly with the needs of society. Quantum algorithms have faster and more efficient capabilities than classic computers. Based on this, designing an efficient quantum algorithm determined by the characteristics of quantum computing has become an important research direction.

We describe quantum walking as: a particle whose walking direction is determined by an evolution operator has two degrees of freedom, and the wave function evolution will occur during its walking. The evolution matrix can be regarded as composed of two matrices: the spin matrix C matrix acting on the coin space and the S matrix which changes the direction of the particles after the spin. The evolution of the spin matrix is ​​only determined by a parameter λ, when we When this parameter is kept constant, the walk behaves as a uniform quantum walk. The difference between a quantum walk with a barrier and a uniform quantum walk is that the conditional shift operator S changes with time or space, and the two parameters that determine the S operator are α and β, respectively., where α is the amplitude at which the particle successfully tunnels and changes its spin state, and β is the amplitude that remains unchanged, that is, the state and spin state of the walker do not change after one step of action. The evolution operator of quantum walk with barriers can be expressed as . A complete step of quantum walking with a uniform barrier is that the particle first performs a spin on its spin state under the action of the C operator, and secondly moves under the action of the S operator. Among them, α and β do not change with time. For quantum walks with non-uniform barriers, α and β change with time.

This paper mainly studies the wave packet diffusion behavior of quantum walking with non-uniform barriers by generating S operators through two types of generalized Fibonacci sequences and two unordered sequences through numerical calculation methods. The parameter αβ changes with time. The probability distribution image of wave packet diffusion is derived by calculation, and the power exponent is fitted to the standard variance, and compared and compared. From the first physical quantity probability distribution p, we find that under the first type of Fibonacci quasi-periodic sequence, the particle wave packet diffusion probability distribution is similar to classical random walking; under the second type of Fibonacci quasi-periodic sequence, the particle wave packet diffusion probability The distribution is similar to the uniform quantum walk. In the case of binary disorder, the probability distribution of particle wave packet diffusion is also similar to uniform quantum walking; in the case of multiple disorder, it is similar to classical random walking. From another point of view, the standard deviation of physical quantities changes with time. Under the first type of Fibonacci quasi-periodic sequence, the wave packet diffusion is sub-ballistic diffusion; under the second type of Fibonacci quasi-periodic sequence, the wave packet diffusion is ballistic diffusion. In the case of binary disorder, particle wave packet diffusion is close to ballistic diffusion; while in the case of multiple disorder, it is close to classical random walking and tends to normal diffusion.

Keywords: quantum random walk; non-uniform barrier; disordered barrier; sub-ballistic diffusion; wave packet diffus

目 录

摘 要 I

Abstract II

第一章 引言 1

1.1 经典随机行走模型 1

1.2 量子行走模型 2

1.3 具有均匀势垒的量子行走模型 5

1.4 广义Fibonacci准周期结构与无序 6

1.5 本选题的研究内容 8

第二章 广义Fibonacci准周期序列生成的具有非均匀势垒的量子行走 9

2.1 引言 9

2.2 第一类序列 9

2.3 第二类广义Fibonacci准周期序列 14

2.4 本章小结 15

第三章 具有无序势垒量子行走波包扩散的动力学行为研究 18

3.1 引言 18

3.2 二元无序 18

3.3 多元无序 20

3.4 本章小结 21

第四章 总结与展望 22

参考文献 23

致谢 24

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