臺灣博碩士論文加值系統

量子電腦的計算加速特性與量子間的糾纏態有關，越多量子比特糾纏，量子計算的加速就越快。因此在量子計算中，量子比特間的糾纏特性穩不穩定就顯得十分重要。真實量子電腦中，會因為許多因素導致量子比特間糾纏狀態不佳，因此，我們就必須藉由一些量子電腦中的運算與測量，來找出哪些量子比特是最適合做計算的。
本研究的第一個計算──Mermin 多項式[1] 對於量子電腦測量的影響，旨在在量子比特間，使用特殊條件類 Greenberger-Horne-Zeilinger 態（GHZlike state；類 GHZ 態）為依據，[2] 並計算 Mermin 多項式，即可以推算量子位元是否具糾纏特性。[3] 第二個計算是 Mermin-Peres 魔方遊戲（MerminPeres Magic Square Game），為一種典型的資訊不對稱遊戲。假設有兩人正在遊玩，除了需滿足 Mermin-Peres 魔方遊戲設下的限制條件外，兩人在遊戲進行過程中全程不能交換訊息。在古典計算中我們可以發現，若要維持 100%勝率，會導致九宮格右下的測量值呈現 1 及 -1。因此，在古典理論中，僅能測出 8/9 的勝率；反之，若是利用量子電腦糾纏的特性預測，將會達到 100%的勝率。計算後得出的結果，每行與每列的機率值大致接近 100%。此結果與量子比特的組合及糾纏狀態是否良好有關。[4]
在本篇中，運用目前兩臺 IBM 5 量子比特的量子電腦── Belem 與Manila 計算並討論 Mermin-Peres 魔方遊戲（Mermin-Peres Magic Square Game）。

The computing acceleration of quantum computer is related to the entanglement state between qubits. The more qubits are entangled the faster the acceleration of quantum computing. Therefore, in quantum computing, it is very important that the entanglement between qubits are stable or not. In a real quantum computer, the entanglement state between qubits will become weak due to many factors. Therefore, we must use some calculations and measurements in quantum computers to find out which qubits are most suitable for calculation.
The first part of discussion is the influence of Mermin polynomial [1] on quantum computer measurement. It bases on the Greenberger-Horne-Zeilinger like state (GHZlike state) between qubits and calculates the Mermin polynomial.[2]. It can be calculated the entanglement properties between qubits[3]. The second part is the Mermin-Peres Magic Square Game, which is a typical information asymmetric game. Assuming that two people are playing the game, the restrictions set by the MerminPeres Magic Square game, the two players cannot exchange messages during the game. In classical calculations, we can calculate the winning rate of the game. In the classical theory, only an 8/9 winning rate can be measured. On the contrary, if the entanglement characteristics of the quantum computer are used to predict the winning rate, it will reach 100%. As a result of the calculation, the probability value of each row and each column is roughly close to 100%. This result is based on the combination of qubits and the states of entanglement are good [4].
At this research, we used the two of IBM 5-qubit quantum computers──Belem and Manila to calculate and discuss the possibility of Mermin-Peres Magic Square Game.

目 錄
書名頁. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
論文口試委審定書 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
中文摘要 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
英文摘要 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
致謝 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
目錄 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
圖目錄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
表目錄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
第一 章 緒論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 前言 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 量子現象與實驗 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 黑體輻射 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 楊格雙狹縫實驗 . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 薛丁格的貓 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4 EPR 悖論 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 實驗簡介 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
第二 章 量子電腦. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 發展歷史 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 現有硬體介紹 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
第三 章 實驗理論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 Pauli X,Y,Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 GHZ-like state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Mermin inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3.1 簡介 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3.2 原理 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.3 量子電路設置 . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Mermin-Peres Magic Square . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4.1 簡介 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4.2 原理 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4.3 量子電路設置 . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 IBM-Belem & IBM-Manila 量子電腦機器線路圖 . . . . . . . . . . . . 28
第四 章 結果討論：Mermin inquality 對於量子電腦測量的影響 . . . . . . . . . 29
4.1 2-5 量子比特在 Qasm Simulator 上的模擬結果 . . . . . . . . . . . . . 29
4.2 真實量子電腦測量結果 . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.1 IBM-Belem 量子電腦的 2-qubit 測量結果 . . . . . . . . . . . . 31
4.2.2 IBM-Manila 量子電腦的 2-qubit 測量結果 . . . . . . . . . . . . 34
4.2.3 IBM-Belem 量子電腦的 3-qubit 測量結果 . . . . . . . . . . . . 37
4.2.4 IBM-Manila 量子電腦的 3-qubit 測量結果 . . . . . . . . . . . . 40
4.2.5 總結 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
第五 章 結果討論：Mermin-Peres 魔方遊戲. . . . . . . . . . . . . . . . . . . . . . 44
5.1 Qasm Simulator 測量結果 . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2 真實量子電腦測量結果 . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2.1 IBM-Belem 量子電腦測量 Mermin-Peres 魔方遊戲結果 . . . . 45
5.2.2 IBM-Manila 量子電腦測量 Mermin-Peres 魔方遊戲結果 . . . . 46
5.2.3 總結 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 增加初始條件 Rz 後 Qasm Simulator 測量結果 . . . . . . . . . . . . . 48
5.4 增加初始條件 Rx 後 Qasm Simulator 測量結果 . . . . . . . . . . . . . 57
第六 章 結論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
參考文獻 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
附錄一. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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