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研究生：	徐嘉成 Xu, Jia-Cheng
論文名稱：	量化單向量子計算及其應用 Quantifying One-Way Quantum Computation and its Applications
指導教授：	李哲明 Li, Che-Ming
學位類別：	碩士 Master
系所名稱：	工學院 - 工程科學系 Department of Engineering Science
論文出版年：	2020
畢業學年度：	108
語文別：	英文
論文頁數：	92
中文關鍵詞：	量子資訊處理、量子計算、量子糾纏、古典過程、量化量子過程
外文關鍵詞：	Quantum information processing, Quantum computing, Quantum entanglement, Classical process, Quantifying quantum processes
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量子計算利用量子力學中特有的性質，提供了效率遠超越古典計算理論之嶄
新資訊處理方法;主要的操作模式分為，量子電路模型以及單向量子計算 (One-Way Quantum Computation)。與前者相比，後者僅需對事先準備好的糾纏態之每個量子位元做特定測量，即實現量子計算結果;其避免了電路模型中，多個量子位元之量子交互問題，為通用的量子計算，提供更直接、可擴展的模型。儘管存在者許多理論和實驗研究，對於量化單向量子計算性能的方法仍不明朗，在更基本的層面，尚不清楚古典的方式可以模擬何種程度的量子計算;單向量子計算利用簇態(Cluster state)作為資源實現算，若以古典實在論描述量子計算中的資源及計算過程，所能展現出最大古典模仿能力為何?實現單向量子計算的過程中，簇態可能因環境干擾或實驗上不可預期之因素，導致喪失其量子特性且能被古典實在論描述，進而影響量子計算的過程及結果;因此量化資源品質與其經歷計算任務之過程中間的關係成為了一個很重要的課題。在本篇論文裡，我們透過使用新型古典模型來模擬計算過程，提出一個量化單向量子計算的方法。利用這項方法，我們可以識別計算結果是否真正經由量子效應所得;更進一步從資源的角度，判斷出簇態喪失其量子特性之型態;摒除掉所有古典模仿的策略，對單向量子計算之過程的可靠性提供一個新的指標。這樣的量化方法，也能直接應用來確認分散式量子計算方案中的信賴性，從計算過程及古典欺騙手法的觀點，提出其他協定中無法做到的信賴性指標，例如盲量子計算、只需量測的盲量子計算。

Quantum computation relies on the peculiar features of quantum mechanics to provide a new information processing method with efficiency far exceeding the classical computation theory; the main modes are divided into the circuit models and oneway quantum computation model. Compared with the former, the latter only needs to make specific measurements for each qubit of the entangled state, that is, to realize the quantum calculation result; it avoids the quantum interaction problem of multiple qubits in the circuit model, providing a more direct and scalable model. Despite the existence of many theoretical and experimental studies, the method of quantifying the performance of oneway quantum computation is still unclear. At a more basic level, it is unclear to what extent classical methods can simulate quantum computation. Oneway quantum computation utilizes cluster states as a resource for realizing computation; if classical realism is used to describe the resources and computational processes in quantum computation, what is the greatest classical imitation capability? In the process of realizing oneway quantum computation, the cluster state may be affected by environmental interference or unpredictable factors in the experiment, lead to the loss of its quantum properties and can be described by classical realism, which in turn affects the process and results of quantum computation; therefore, quantifying the correlation between the quality of resources and the process of undergoing tasks has become an important issue. In this paper, we use a novel classical model to simulate the computation process and propose a method for quantifying oneway quantum computation. Utilizing this method, we can identify whether the calculation result is truly obtained through quantum effects; further, from the viewpoint of resources, classify the case of cluster state that loses its quantum characteristics; eliminate all classical imitation strategies and provide a new indicator for the reliability of the process of oneway quantum computation. Such a quantify method can also be directly applied to confirm the reliability of the distributed quantum computation protocols. From the viewpoint of the computation process and classical deception methods, it proposes reliability indicators that cannot be achieved in other protocols, such as blind quantum computation, measurementonly blind quantum computation.

摘要                                               i
Abstract                                          ii
誌謝                                              iv
Table of Contents                                  v
List of Tables                                   viii
List of Figures                                   ix
Nomenclature                                      xi

Chapter 1. Introduction 1
1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3. Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Chapter 2. Essential Knowledge and Tools 7
1. Postulates of quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1. Postulate 1 – State space . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2. Postulate 2 – Quantum evolution . . . . . . . . . . . . . . . . . . . . . . . 9
1.3. Postulate 3 – Quantum measurement . . . . . . . . . . . . . . . . . . . . . . 10
1.4. Postulate 4 – Composite system . . . . . . . . . . . . . . . . . . . . . . . 12
2. The density operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3. Quantum tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1. Quantum state tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2. Quantum process tomography . . . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 3. Introduction to OneWay Quantum Computation Model and Its Applications 21
1. Quantum computation: circuit model and one-way model . . . . . . . . . . . . . 22
2. One-way quantum computation model . . . . . . . . . . . . . . . . . . . . . . . 24
2.1. Cluster states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2. One-way quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3. Four-qubit cluster states for realizing universal quantum computer  . . 30
2.4. Bell-state measurement for realizing universal quantum computer  . . . . 31
3. Secure delegated quantum computations . . . . . . . . . . . . . . . . . . . . . 37
3.1. Blind quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2. The security in blind quantum computation . . . . . . . . . . . . . . . . . . 39
3.3. Measurement-only blind quantum computation . . . . . . . . . . . . . . . . . 42
3.4. The security in measurement-only blind quantum computation . . . . . 43

Chapter 4. Quantifying One-Way Quantum Computation 48
1. Basic idea of classical computation . . . . . . . . . . . . . . . . . . . . . . 48
2. Classical one-way computation model . . . . . . . . . . . . . . . . . . . . . . 49
2.1. Introduction to classical one-way computation model . . . . . . . . . . . . . 50
2.2. Characterizing one-way quantum computation processes . . . . . . . . . . 52
2.3. General model of classical one-way computation . . . . . . . . . . . . . . . 54
3. Identification and quantification of one-way quantum computation . . . . . 60
3.1. Introduction to semidefinite programming . . . . . . . . . . . . . . . . . . 60
3.2. Fidelity criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3. Quantum composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4. Quantum robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4. Example of quantifying one-way quantum computation . . . . . . . . . . . . . . 71
5. Comparison of our framework with others in identification and reliability . 78
6. Discussion and Applications . . . . . . . . . . . . . . . . . . . . . . . . ..  82
6.1. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.2. Applications in secure delegated quantum computations . . . . . . . . . . . . 82

Chapter 5. Summary and Outlook 85
1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

References 88
                                    

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