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In this paper, a feedback sum-product decoding algorithm of sparse quantum codes for X-Z Pauli channels is developed. Compared with the previous decoding algorithm, our feedback strategy exploits not just the syndrome but also the values of the frustrated checks on individual qubits of the code and the character of the channel model with the portion of each error to adjust the probability distribution of information nodes. Due to the smart adjustment, our decoding algorithm, on one hand, can break the symmetric degeneracy, and on the other hand, can feed back more useful information to the SPA decoder to help the decoder determine a valid output, thereby significantly improving the decoding ability of the decoder. Moreover, our algorithm, which is based on GF(4), overcomes the limitation caused by decoding in GF(2). Finally, we want to point out that, our method does not increase the measurement overhead in comparison wioth the previous methods, as the extra information comes for free from the requisite stabilizer measurement.
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Keywords:
- sparse quantum codes /
- sum-product algorithm /
- quantum error-correcting codes /
- quantum information
[1] Liu W J, Chen H W, Ma T H, Li Z Q, Liu Z H, Hu W B 2009 Chin. Phys. B 18 4105
[2] Wang X B 2004 Phys. Rev. Lett. 92 077902
[3] Li C Y, Li X H, Deng F G, Zhou H Y 2008 Chin. Phys. B 17 2352
[4] Zhou N C, Zeng G H, Gong L H, Liu S Q 2007 Acta Phys. Sin. 56 9 (in Chinese) [周南润、曾贵华、龚黎秋、刘三秋2007 56 9]
[5] Zhang S, Wang J, Zhang Q, Tang C J 2009 Acta Phys. Sin. 58 1 (in Chinese)[张 盛、 王 剑、张 权、唐朝京 2009 58 1]
[6] Gallager R G 1962 IRE Trans. Inform. Theory 8 21
[7] Li Z, Xing L J 2008 Acta Phys. Sin. 57 28 (in Chinese) [李 卓、 邢莉娟 2008 57 28]
[8] MacKay D J C, Mitchison G J, McFadden P L 2004 IEEE Trans. Inform. Theory 50 2315
[9] Wang Y J, Bai B M, Zhao W B, Wang X M 2009 Int. J. Quantum Inf. 7 1373
[10] Li Y, Zeng G H, Moon H L 2009 Chin. Phys. B 18 4154
[11] Poulin D, Chung Y 2008 Quantum Inform. Comput. 8 987
[12] Wang Y J, Sanders B C, Bai B M, Wang X M 2009 Proceedings of the 9th Asian Conference on Quantum Information Science Nanjing, P.R. China, August 2009 p79
[13] Gottesman D 1996 Phys. Rev. A 54 1862
[14] Camara T, Ollivier H, Tillich J P 2007 Proceedings of the International Symposium on Information Theory Nice, France, June 2007 p811
[15] Steane A M 1996 Phys. Rev. Lett. 77 793
[16] Calderbank A R, Shor P W 1996 Phys. Rev. A 54 1098
[17] Preskill J 2001 Lecture Notes for Physics 219: Quantum Computation. Chap 7 p5
[18] Wang Y J, Sanders B, Bai B M, Wang X M 2009 arxiv.org/abs/0912.4546
[19] Calderbank A R, Rains E M, Shor P W, Sloane N J A 1997 Phys. Rev. Lett. 78 405
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[1] Liu W J, Chen H W, Ma T H, Li Z Q, Liu Z H, Hu W B 2009 Chin. Phys. B 18 4105
[2] Wang X B 2004 Phys. Rev. Lett. 92 077902
[3] Li C Y, Li X H, Deng F G, Zhou H Y 2008 Chin. Phys. B 17 2352
[4] Zhou N C, Zeng G H, Gong L H, Liu S Q 2007 Acta Phys. Sin. 56 9 (in Chinese) [周南润、曾贵华、龚黎秋、刘三秋2007 56 9]
[5] Zhang S, Wang J, Zhang Q, Tang C J 2009 Acta Phys. Sin. 58 1 (in Chinese)[张 盛、 王 剑、张 权、唐朝京 2009 58 1]
[6] Gallager R G 1962 IRE Trans. Inform. Theory 8 21
[7] Li Z, Xing L J 2008 Acta Phys. Sin. 57 28 (in Chinese) [李 卓、 邢莉娟 2008 57 28]
[8] MacKay D J C, Mitchison G J, McFadden P L 2004 IEEE Trans. Inform. Theory 50 2315
[9] Wang Y J, Bai B M, Zhao W B, Wang X M 2009 Int. J. Quantum Inf. 7 1373
[10] Li Y, Zeng G H, Moon H L 2009 Chin. Phys. B 18 4154
[11] Poulin D, Chung Y 2008 Quantum Inform. Comput. 8 987
[12] Wang Y J, Sanders B C, Bai B M, Wang X M 2009 Proceedings of the 9th Asian Conference on Quantum Information Science Nanjing, P.R. China, August 2009 p79
[13] Gottesman D 1996 Phys. Rev. A 54 1862
[14] Camara T, Ollivier H, Tillich J P 2007 Proceedings of the International Symposium on Information Theory Nice, France, June 2007 p811
[15] Steane A M 1996 Phys. Rev. Lett. 77 793
[16] Calderbank A R, Shor P W 1996 Phys. Rev. A 54 1098
[17] Preskill J 2001 Lecture Notes for Physics 219: Quantum Computation. Chap 7 p5
[18] Wang Y J, Sanders B, Bai B M, Wang X M 2009 arxiv.org/abs/0912.4546
[19] Calderbank A R, Rains E M, Shor P W, Sloane N J A 1997 Phys. Rev. Lett. 78 405
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