近十年來，因量子電腦建構方法的提出，量子領域的研究開始蓬勃地發展，為了解決量子態會因環境的影響產生相消干(decoherence)的問題，以及在量子通訊上通道的影響，如何更正量子位元的錯誤，是量子錯誤更正碼研究的主要目標。其中量子穩定碼是最為廣泛研究的量子錯誤更正碼，因為量子穩定碼可藉由古典的同位元檢查矩陣來建構，並藉由錯誤徵狀來偵測並修正量子位元錯誤，和古典的線性碼有很多類似的性質。然而，量子穩定碼建構相關的檢查矩陣須滿足交換關係，使得量子穩定碼一直未能有較CSS建構法更簡單的建構方式。

在本篇論文裡，我們針對量子穩定碼的檢查矩陣建構法，提出如何簡單滿足交換關係的建構方式。首先，我們提出使用古典循環碼的建構法，藉由生成矩陣的一些變化，可以簡單地滿足交換關係，但是量子最小碼距，仍未有方法使其達到最大限度。其次，我們研究使用二元二次剩餘碼的特性來建構的兩種量子穩定碼－CSS建構法和量子二元二次剩餘碼，比較兩種建構法下其量子穩定碼的特性，包括適用碼長、最小碼距。藉由量子二元二次剩餘碼的啟發，[[n, 1]]量子穩定碼已被發現可藉由某些指引向量簡單地建構出來，我們針對指引向量，提出一個建構準則和相關的性質。接著我們將該種建構法延伸發展，提出[[n, k]]量子穩定碼的簡單建構方式，並指出該量子穩定碼最小碼距和指引向量之間的關係。但如何使得建構的量子穩定碼可以達到最小碼距的最大限距，則是未來可以思考及研究的方向。

In the study of quantum error-correcting codes, stabilizer codes is perhaps the most
important ones and can be constructed by classical self-orthogonal codes. In this thesis,
our aim is to pursue a further study of the structure of quantum stabilizer codes based
on syndrome assignment by classical parity check matrices. We proposed a construction
by using two cyclic codes. We also found the normalizer of quantum quadratic-residue
codes, which helps nd the minimum distance. Finally, a construction of [[n; k]] quantum
stabilizer codes with k > 1 was proposed.

第一章 簡介
第二章 量子穩定碼的建構
第三章 基於二元二次剩餘碼的量子穩定碼建構
第四章 二元二次剩餘碼啟發的量子穩定碼建構
第五章 結論
附　錄　 英文論文本

Contents
1 Introduction 1
2 Construction of Quantum Stabilizer Codes 3
2.1 Stabilizer Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Binary Representation of Stabilizer Groups . . . . . . . . . . . . . . . . . 4
2.3 Construction of Quantum Stabilizer Codes . . . . . . . . . . . . . . . . . 6
2.3.1 Error Syndromes . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.2 Syndrome Assignment by a Binary Parity-Check Matrix . . . . . 7
2.3.3 Some Construction of Check Matrices . . . . . . . . . . . . . . . . 8
3 Construction of Quantum Stabilizer Codes by Binary Quadratic-Residue
Codes 11
3.1 Quantum Circulant Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Quadratic-Residue Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 CSS Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4 Quantum Quadratic-Residue Codes . . . . . . . . . . . . . . . . . . . . . 14
3.5 Comparison About Quantum Minimum Distance . . . . . . . . . . . . . 16
4 Construction of Quantum Codes Inspired by Binary Quadratic-Residue
Codes 18
4.1 A Construction for Quantum Codes with k = 1 . . . . . . . . . . . . . . 18
4.2 A Construction for Quantum Codes with k = 2 . . . . . . . . . . . . . . 21
4.3 A General Construction for Quantum Codes . . . . . . . . . . . . . . . . 25
5 Conclusion 28

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