Concatenated Quantum Codes Achieving High Rates With Polynomial-Time Error Estimation or Construction

A method for concatenating quantum errorcorrecting codes is presented. The method is applicable to a wide class of quantum error-correcting codes known as Calderban kShor-Steane (CSS) codes. As a result, codes that achieve a hi gh rate in the Shannon theoretic sense and that are decodable in polynomial time are presented. The rate is the highest among those known to be achievable by CSS codes. Moreover, the know n bound best on the minimum distance of codes constructible in polynomial time is improved for a wide range.

[1]  Shor,et al.  Simple proof of security of the BB84 quantum key distribution protocol , 2000, Physical review letters.

[2]  I. Devetak,et al.  The private classical information capacity and quantum information capacity of a quantum channel , 2003 .

[3]  A. Steane.  Multiple-particle interference and quantum error correction , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[4]  A. Calderbank,et al.  Quantum Error Correction and Orthogonal Geometry , 1996, quant-ph/9605005.

[5]  Gérard D. Cohen,et al.  On binary constructions of quantum codes , 1999, IEEE Trans. Inf. Theory.

[6]  David J. C. MacKay,et al.  Sparse-graph codes for quantum error correction , 2004, IEEE Transactions on Information Theory.

[7]  N. Sloane,et al.  Quantum error correction via codes over GF(4) , 1996, Proceedings of IEEE International Symposium on Information Theory.

[8]  S. Litsyn,et al.  On binary constructions of quantum codes , 1998, Proceedings of the 1999 IEEE Information Theory and Communications Workshop (Cat. No. 99EX253).

[9]  Shor,et al.  Good quantum error-correcting codes exist. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[10]  Kenneth W. Shum,et al.  A low-complexity algorithm for the construction of algebraic-geometric codes better than the Gilbert-Varshamov bound , 2001, IEEE Trans. Inf. Theory.

[11]  Gottesman.  Class of quantum error-correcting codes saturating the quantum Hamming bound. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[12]  Ryutaroh Matsumoto,et al.  Improvement of Ashikhmin-Litsyn-Tsfasman bound for quantum codes , 2002, Proceedings IEEE International Symposium on Information Theory,.

[13]  A. D. Wyner,et al.  The wire-tap channel , 1975, The Bell System Technical Journal.

[14]  Chaoping Xing,et al.  Asymptotic bounds on quantum codes from algebraic geometry codes , 2006, IEEE Transactions on Information Theory.

[15]  M. Hamada.  Information rates achievable with algebraic codes on quantum discrete memoryless channels , 2005, IEEE Transactions on Information Theory.

[16]  H. Stichtenoth.  Transitive and Self-Dual Codes Attaining the , 2006 .

[17]  Alexei E. Ashikhmin,et al.  Nonbinary quantum stabilizer codes , 2001, IEEE Trans. Inf. Theory.

[18]  Andrew M. Steane.  Enlargement of Calderbank-Shor-Steane quantum codes , 1999, IEEE Trans. Inf. Theory.

[19]  S. Litsyn,et al.  Asymptotically Good Quantum Codes , 2000, quant-ph/0006061.

[20]  Mitsuru Hamada,et al.  Conjugate codes and applications to cryptography , 2006 .

[21]  Tom Høholdt,et al.  An explicit construction of a sequence of codes attaining the Tsfasman-Vladut-Zink bound: The first steps , 1997, IEEE Trans. Inf. Theory.

[22]  Abraham Lempel,et al.  Factorization of Symmetric Matrices and Trace-Orthogonal Bases in Finite Fields , 1980, SIAM J. Comput..

[23]  Hao Chen,et al.  Asymptotically good quantum codes exceeding the Ashikhmin-Litsyn-Tsfasman bound , 2001, IEEE Trans. Inf. Theory.

[24]  R. Cleve,et al.  Efficient computations of encodings for quantum error correction , 1996, quant-ph/9607030.

[25]  Mitsuru Hamada,et al.  Constructive Conjugate Codes for Quantum Error Correction and Cryptography , 2007, ArXiv.

[26]  Mitsuru Hamada,et al.  Quotient Codes and Their Reliability , 2005 .

[27]  Fujita Hachiro.  Several Classes of Concatenated Quantum Codes: Constructions and Bounds , 2006 .

[28]  H. Stichtenoth,et al.  On the Asymptotic Behaviour of Some Towers of Function Fields over Finite Fields , 1996 .

[29]  Mitsuru Hamada,et al.  Reliability of Calderbank?Shor?Steane codes and security of quantum key distribution , 2004 .

[30]  Douglas A. Leonard,et al.  Finding the defining functions for one-point algebraic-geometry codes , 2001, IEEE Trans. Inf. Theory.