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Mapping the error syndromes to error operators is the core of quantum decoding network and the key step to realize quantum error correction. The definitions of the bit flip error syndrome matrix and the phase flip error syndrome matrix are presented, and then the error syndromes of Pauli errors are expressed in terms of the columns of the bit flip error syndrome matrix and the phase flip error syndrome matrix. It is also shown that the error syndrome matrix of a stabilizer code is determined by its check matrix, which is similar to the relationship between the classical error and the parity check matrix of classical codes. So, the techniques of error detection and error correction for classical linear codes can be applied to quantum stabilizer codes after some modifications. The error correction circuits are constructed based on the relationship between the error operator and error syndrom. The decoding circuit is constructed by reversing the encoding circuit because the encoding operators are unitary.
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Keywords:
- stabilizer code /
- check matrix /
- error syndrome /
- Pauli operator
[1] Nielsen M A, Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) p72
[2] [3] Calderbank A R, Rains E M, Shor P W, Sloane J A 1998 IEEE Trans. Inform. Theory 44 1369
[4] Ketkar A, Klappenecker A, Kumar S 2006 IEEE Trans. Inform. Theory 52 4892
[5] [6] [7] Li Y, Zeng G H, Moon H L 2009 Chin. Phys. B 18 4154
[8] [9] Cleve R, Gottesman D 1997 Phys. Rev. A 56 76
[10] Gottesman D 1997 Ph. D. Dissertation (Pasadena: California Institute of Technology)
[11] [12] Wu C H, Tsai Y C, Tsai H L 2005 Circuits and Systems (Kobo: Springer-Verlag) p23
[13] [14] [15] Forney G D, Grassl M, Guha S 2007 IEEE Trans. Inform. Theory 53 865
[16] [17] Wilde M M 2009 Phys. Rev. A 79 062325
[18] Poulin D, Chung Y J 2008 Quantum Inform. Comput. 8 987
[19] [20] [21] Evans Z W E, Stephens A M 2008 Phys. Rev. A 78 062317
[22] [23] Poulin D, Tillich J P 2009 IEEE Trans. Inform. Theory 55 2776
[24] Li Z, Xing L J, Wang X M 2008 J. Xidian Univ.(Nat. Sci. Ed.) 35 834 (in Chinese) [李 卓、 邢莉娟、 王新梅 2008 西安电子科技大学学报(自然科学版) 35 834]
[25] [26] [27] Xing L J, Li Z, Bai B M, Wang X M 2008 Acta Phys. Sin. 57 4695 (in Chinese) [邢莉娟、 李 卓、 白宝明、 王新梅 2008 物理学报 57 4695]
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[1] Nielsen M A, Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) p72
[2] [3] Calderbank A R, Rains E M, Shor P W, Sloane J A 1998 IEEE Trans. Inform. Theory 44 1369
[4] Ketkar A, Klappenecker A, Kumar S 2006 IEEE Trans. Inform. Theory 52 4892
[5] [6] [7] Li Y, Zeng G H, Moon H L 2009 Chin. Phys. B 18 4154
[8] [9] Cleve R, Gottesman D 1997 Phys. Rev. A 56 76
[10] Gottesman D 1997 Ph. D. Dissertation (Pasadena: California Institute of Technology)
[11] [12] Wu C H, Tsai Y C, Tsai H L 2005 Circuits and Systems (Kobo: Springer-Verlag) p23
[13] [14] [15] Forney G D, Grassl M, Guha S 2007 IEEE Trans. Inform. Theory 53 865
[16] [17] Wilde M M 2009 Phys. Rev. A 79 062325
[18] Poulin D, Chung Y J 2008 Quantum Inform. Comput. 8 987
[19] [20] [21] Evans Z W E, Stephens A M 2008 Phys. Rev. A 78 062317
[22] [23] Poulin D, Tillich J P 2009 IEEE Trans. Inform. Theory 55 2776
[24] Li Z, Xing L J, Wang X M 2008 J. Xidian Univ.(Nat. Sci. Ed.) 35 834 (in Chinese) [李 卓、 邢莉娟、 王新梅 2008 西安电子科技大学学报(自然科学版) 35 834]
[25] [26] [27] Xing L J, Li Z, Bai B M, Wang X M 2008 Acta Phys. Sin. 57 4695 (in Chinese) [邢莉娟、 李 卓、 白宝明、 王新梅 2008 物理学报 57 4695]
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