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.

[1] W. K. Wootters and W. H. Zurek, "A single quantum cannot be cloned," Nature,
vol. 299, no. 5886, pp. 802-803, 1982.
[2] P. W. Shor, "Scheme for reducing decoherence in quantum computer memory,"
Phys. Rev. A, vol. 52, p. 2493, 1995.
[3] A. M. Steane, "Error correcting codes in quantum theory," Phys. Rev. Lett., vol. 77,
no. 5, pp. 793-797, 1996.
[4] A. R. Calderbank and P. W. Shor, "Good quantum error-correcting codes exist,"
Phys. Rev. A, vol. 54, no. 2, pp. 1098-1106, 1996.
[5] A. M. Steane, "Simple quantum error correcting codes," Phys. Rev. Lett., vol. 77,
pp. 793-797, 1996.
[6] A. Steane, "Multiple particle interference and quantum error correction," Proc.
Roy. Soc. Lond. A, vol. 452, pp. 2551-2576, 1996.
[7] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, "Quantum error
correction and orthogonal geometry," Phys. Rev. Lett., vol. 78, no. 3, pp. 405-408,
1997.
[8] D. Gottesman, "Class of quantum error-correcting codes saturating the quantum
hamming bound," Phys. Rev. A, vol. 54, no. 3, pp. 1862-1868, 1996.
[9] D. Gottesman, "Stabilizer codes and quantum error correction," Ph.D. dissertation, Califor-
nia Institute of Technology, Pasadena, CA, 1997.
[10] C. H. Bennett, D. DiVincenzo, J. A. Smolin, and W. K. Wootters, "Mixed state
entanglement an quantum error correction," Phys. Rev. A, vol. 54, pp. 3824-3851,
1996.
[11] A. Ekert and C. Macchiavello, "Error correction in quantum communication," Phys.
Rev. Lett., vol. 77, pp. 2585-2588, 1996.
[12] E. Knill and R. La
amme, "Theory of quantum error-correcting codes," Phys. Rev.
A, vol. 55, pp. 900-911, 1997.
[13] C.-Y. Lai and C.-C. Lu, "A construction of quantum stabilizer codes based on
syndrome assignment by classical parity-check matrices," arXiv:0712.0103v1[quant-
ph].
[14] A. R. Calderbank, E. Rains, P. W. Shor, and N. J. A. Sloane, "Quantum error
correction via codes over GF(4)," IEEE Trans. Inform. Theory, vol. 44, no. 4, pp.
1369-1387, 1998.
[15] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes.
Amdterdam, The Netherlands: North-Holland, 1977.
[16] A. M. Steane, "Enlargement of calderbank shor steane quantum codes," IEEE
Trans. Inform. Theory, vol. 45, no. 7, pp. 2492-2495, 1999.
[17] T.-K. Truong, Y. Chang, and C.-D. Lee, "The weight distributions of some binary
quadratic residue codes," IEEE Trans. Inform. Theory, vol. 51, no. 5, pp. 1776-
1782, 2005.