论文总字数：26921字

摘 要

纠缠是量子的特有属性，在量子计算、量子通信中被广泛应用，被认为是使量子算法相比于传统算法更强大的主要原因之一。但关于量子纠缠的现有研究主要集中量子纠缠的测度，以及利用量子纠缠的特性实现通信等方面，忽略了量子比特个数对其的影响。在本论文中，我们将讨论量子态纠缠度的分析方法——纠缠测度法，并对不同量子比特数下的量子态做了纠缠度计算分析，同时使用纠缠测度对单目标和多目标的Grover搜索算法产生的量子纠缠进行分析，尝试探究量子纠缠对量子算法的影响。

量子态是量子算法中的关键部分（GHZ态是量子态中一种特殊态），而纠缠度是体现量子态特殊性的重要特征。分析量子态纠缠度的主要方法是使用纠缠测度，纠缠测度的有不同的定义方法，其中比较常用的是Groverian测度和几何测度。在本文中Groverian测度是作为分析的主要手段，也会适当使用几何测度加以比较，因为在我们实验研究进行过程中主要涉及的是纯态，故选择Groverian测度和几何测度均选择适用于纯态的定义方式。具体来说，我们的主要工作以及其结果如下：

其一、我们使用分块法一定程度上简化了Groverian测度的计算方法，并仿真计算了一定范围内GHZ态的Groverian测度，得出该范围内GHZ态的Groverian测度相等的结论，然后我们通过数学推导也得出了同样的结论。进一步的推广到广义GHZ态上，我们从仿真计算和理论推导两个方面得出了同一类广义GHZ态具有一样的Groverian测度。使用几何测度对GHZ态和广义GHZ态进行分析，也得出了同样的结论。

其二、我们使用Groverian测度和几何测度分别对单目标和多目标Grover搜索算法演算过程中的量子态的纠缠度进行分析，尝试分析纠缠度与算法有效性之间的关系。当搜索算法是单目标时，我们发现量子态的纠缠度先增大后变小，纠缠度的增大使得算法变高效，而之后纠缠度的变小使得较为准确的测量变得可能。当搜索算法是多目标时，纠缠度变化略有一些不规则性，总体趋势是增加或者先增后减，不能够得出有效结论。我们尝试使用目标数量对纠缠测度公式进行修正，以获得有效的评价算法有效性的方法，不过进一步结论还需要更多的分析。

其三、在Groverian测度和几何测度不断比较的过程中，虽然这两者的计算结果在数值上略有差异，但是却有相同的增减性，并能够导向相同的结果，这也间接性的说明了这两种纠缠测度法在纠缠度分析上的有效性。

总的来说，本文中使用到的核心方法引用于前人的工作，但也提出了自己的一些自己的改进，使用分块法计算Groverian纠缠测度，并使用这些方法分析算法尝试讨论量子纠缠度和算法有效性之间的联系。使用数值和分析两种方法，我们成功的分析了不同量子比特数下的GHZ态的纠缠度。而量子纠缠度和算法有效性联系方面则给出了初步的结论，单目标算法可以使用量子纠缠度作为算法有效性评价手段，但是多目标情况下，单纯的纠缠度不能够成为有效的分析方法，需要配合量子目标数量对公式进行修正。文中给出了修正公式的一种定义方法，但是进一步的结论需要进一步的研究。

关键词

量子计算；纠缠测度；量子算法有效性；Groverian测度；几何测度

Entanglement analysis of GHZ states with different qubits

Student MingYou Wu

Supervised by Hanwu Chen

ABSTRACT

Entanglement is a special property of quantum, and is widely used in quantum computation and quantum communication. It is considered to be one of the main reasons that makes quantum algorithm more powerful than traditional algorithm. However, the existing researches on quantum entanglement mainly focus on the measurement of quantum entanglement, and use the properties of quantum entanglement to realize communication, and ignore the influence of the number of qubits. In this paper, we will discuss the analysis method of quantum entanglement, quantum entanglement measurement, and analyze different number of qubits using quantum entanglement measurement. At the same time, the entanglement measure is used to analyze the quantum entanglement produced by the single target and multi objective Grover search algorithm, and the effect of quantum entanglement on the quantum algorithm is explored.

Quantum states are the key parts of quantum algorithms (GHZ states are a special state in quantum states), and entanglement is an important feature of quantum states. The main method to analyze entanglement of quantum states is to use entanglement measure. There are different definitions of entanglement measure, among which Groverian measure and geometric measure are often used. In this paper, the Groverian measure is used as the primary means of analysis and will be compared with the proper geometric measure. Since the main process involved in our experimental research is pure state, the choice of Groverian measure and geometric measure are all applicable to the definition of pure states. Specifically, our main work and its results are as follows:

First, we use block method to simplify the calculation of Groverian measure to a certain extent. The Groverian measure of the GHZ state in a certain range is simulated and calculated, and the conclusion that the Groverian measure of the GHZ state is equal in this range is obtained, and we draw the same conclusion by mathematical deduction. Further, this conclusion is extended to the generalized GHZ state. From the two aspects of simulation and theoretical deduction, we obtain the same Groverian measure of the same class of generalized GHZ states. Using geometric measure to analyze GHZ state and generalized GHZ state, we also come to the same conclusion.

Second, we analyze the entanglement of the use of Groverian measure and geometric measure of single objective and multi-objective Grover search algorithm of quantum states in the process, try to analyze the relationship between entanglement and the effectiveness of the algorithm. When the search algorithm is a single objective, we find that the entanglement of Quantum States increases first and then decreases. The increase of entanglement makes the algorithm more efficient, and the smaller degree of entanglement makes it possible to make more accurate measurements. When the search algorithm is multi-objective, the degree of entanglement changes slightly irregularly, and the overall trend is increasing, or increasing first and then decreasing, and we can’t get an effective conclusion. We try to use the number of targets to correct the measure of entanglement, in order to obtain an effective method to evaluate the effectiveness of the algorithm. However, further analysis is needed.

Third, in the process of Groverian measure and geometric measure constant comparison, although the two calculation results in a slight difference in value, but has the same change, and can lead to the same result, this also indirectly indicate that two kinds of entanglement measure method in the analysis of the effective entanglement.

In general, the core method used in this article cited in the previous work, but also has made some improvements in their own, using block method to compute Groverian entanglement measure, and using these methods to analyze the relation between quantum entanglement and algorithm validity. By using two methods of numerical analysis and analysis, we successfully analyze the entanglement of GHZ states with different qubit numbers. But the relationship between quantum entanglement and the effectiveness of the algorithm is given a preliminary conclusion, the single objective algorithm can use quantum entanglement as a means of evaluation of the effectiveness of the algorithm, but the multi-objective case, pure entanglement cannot become the effective analysis method, the formula need to be modified with the number of quantum targets. A definition method of the correction formula is given in this paper, but the further conclusion needs further study

KEY WORDS

Quantum Computing；Quantum Entanglement Measure；Quantum Algorithm effectiveness；Groverian Measure；Geometric Measure

目 录

摘 要 I

关键词 II

ABSTRACT III

Keywords IV

第一章 绪论 1

1.1. 引言 1

1.2. 研究背景 2

1.3. 研究意义 3

1.4. 主要贡献 4

1.5. 章节安排 4

第二章 预备知识：Grover算法以及纠缠测度法 4

2.1. Grover算法 4

2.1.1. Oracle 5

2.1.2. 计算过程 5

2.1.3. 算法原理 7

2.1.4. 仿真结果 8

2.2. Groverian测度 10

2.2.1. 任意初始态的Grover搜索算法 10

2.2.2. 搜索成功概率 11

2.2.3. Groverian测度的定义 13

2.2.4. Groverian测度的计算 14

2.2.5. Groverian测度仿真计算时的注意以及改进 15

2.3. 几何测度 16

第三章 GHZ态和广义GHZ态的Groverian测度及其分析 17

3.1. GHZ态的Groverian测度与分析 17

3.2. 广义GHZ态的Groverian测度与分析 18

第四章 GHZ态和广义GHZ态的几何测度 19

4.1. GHZ态的几何测度 19

4.2. 广义GHZ态的几何测度 20

第五章 单目标Grover算法中量子态的测度的变化及其分析 21

5.1. 单搜索目标时量子态的Groverian测度与分析 21

5.2. 单搜索目标时量子态的几何测度与分析 22

5.3. 单目标算法有效性与纠缠测度变化的分析 22

第六章 多目标Grover算法中量子态测度的变化与分析 23

6.1. 多搜索目标时各量子态的Groverian测度与分析 23

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