Research PapersNo Access

Entanglement-assisted quantum codes constructed from primitive quaternary BCH codes

Liang-Dong Lü

School of Science, Air Force Engineering University, Xi'an, Shaanxi 710051, P. R. China

Search for more papers by this author

and

Ruihu Li

School of Science, Air Force Engineering University, Xi'an, Shaanxi 710051, P. R. China

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0219749914500154Cited by:39 (Source: Crossref)

Abstract

The entanglement-assisted (EA) formalism generalizes the standard stabilizer formalism. All quaternary linear codes can be transformed into entanglement-assisted quantum error correcting codes (EAQECCs) under this formalism. In this work, we discuss construction of EAQECCs from Hermitian non-dual containing primitive Bose–Chaudhuri–Hocquenghem (BCH) codes over the Galois field GF(4). By a careful analysis of the cyclotomic cosets contained in the defining set of a given BCH code, we can determine the optimal number of ebits that needed for constructing EAQECC from this BCH code, rather than calculate the optimal number of ebits from its parity check matrix, and derive a formula for the dimension of this BCH code. These results make it possible to specify parameters of the obtained EAQECCs in terms of the design parameters of BCH codes.

Keywords:

References

D. Gottesman, An introduction to quantum error correction, (2000) , arXiv:quant-ph/0004072 v1 . Google Scholar
A. R. Calderbank and P. W. Shor , Phys. Rev. A 54 , 1098 ( 1996 ) . Crossref, Web of Science, Google Scholar
A. M. Steane , Phys. Rev. Lett. 77 , 793 ( 1999 ) . Crossref, Web of Science, Google Scholar
A. R. Calderbank et al. , IEEE Trans. Inf. Theory 44 , 1369 ( 1998 ) . Crossref, Web of Science, Google Scholar
A. Ashikhmin and E. Knill , IEEE Trans. Inf. Theory 47 , 3065 ( 2001 ) . Crossref, Web of Science, Google Scholar
A. Ketkar et al. , IEEE Trans. Inf. Theory 52 , 4892 ( 2006 ) . Crossref, Web of Science, Google Scholar
T. Brun , I. Devetak and M.-H. Hsieh , Science 52 , 436 ( 2006 ) . Crossref, Web of Science, Google Scholar
T. Brun, I. Devetak and M.-H. Hsieh, Catalytic quantum error correction, (2006) , arXiv:quant-ph/0608027v2 . Google Scholar
M.-H. Hsieh , I. Devetak and T. Brun , Phys. Rev. A 76 , 062313 ( 2007 ) . Crossref, Web of Science, Google Scholar
M. Wilde and T. Brun , Phys. Rev. A 77 , 064302 ( 2008 ) . Crossref, Web of Science, Google Scholar
C. Y. Lai and T. Brun, Entanglement increases the error-correcting ability of quantum error-correcting codes, [quant-ph] (2010) , arXiv:1008.2598v2 . Google Scholar
W. C. Huffman and V. Pless , Fundamentals of Error-Correcting Codes ( Cambridge University Press , Cambridge, U.K. , 2003 ) . Crossref, Google Scholar
M. Grassl , T. Beth and T. Pellizzari , Phys. Rev. Lett. A 56 , 33 ( 1997 ) . Crossref, Web of Science, Google Scholar
M. Grassl and T. Beth , Proc. Royal Soc. London Series A 456 , 2689 ( 2000 ) . Crossref, Web of Science, Google Scholar
G. La Guardia , Phys. Rev. A 80 , 042331 ( 2009 ) . Crossref, Web of Science, Google Scholar
S. A. Aly, A. Klappenecker and P. K. Sarvepalli, Primitive quantum BCH codes over finite fields, (2006) , arXiv:quant-ph/0501126v2 . Google Scholar
R. Li , F. Zuo and Y. Liu , J. Air Force Eng. Univ. (Nat. Sci. Ed.) 12 , 87 ( 2011 ) . Google Scholar
A. Steane , IEEE Trans. Inform. Theory 45 , 2492 ( 1999 ) . Crossref, Web of Science, Google Scholar
S. A. Aly , A. Klappenecker and P. K. Sarvepalli , IEEE Trans. Inform. Theory 53 , 1183 ( 2007 ) . Crossref, Web of Science, Google Scholar
R. Li, Research on additive quantum error correcting codes, Ph.D. thesis, Northwest Poly- technical University (2004) . Google Scholar
M. Grassl and T. Beth, Quantum BCH codes, Proc. X. Int. Symp. Theoretical Electrical Engineering (Magdeburg, 1999) pp. 207–212. Google Scholar
R. Li , G. Xu and D. Lu , J. Air Force Eng. Univ. (Nat. Sci. Ed.) 14 , 86 ( 2013 ) . Google Scholar
R. Li et al. , Quantum Inform. Comput. 13 , 0021 ( 2013 ) . Crossref, Web of Science, Google Scholar