Nonbinary quantum error-correcting codes from algebraic curves

We give a new exposition and proof of a generalized CSS construction for nonbinary quantum error-correcting codes. Using this we construct nonbinary quantum stabilizer codes with various lengths, dimensions, and minimum distances from algebraic curves. We also give asymptotically good nonbinary quantum codes from a Garcia-Stichtenoth tower of function fields which are constructible in polynomial time.

[1]  Venkatesan Guruswami,et al.  Improved decoding of Reed-Solomon and algebraic-geometry codes , 1999, IEEE Trans. Inf. Theory.

[2]  M. Tsfasman,et al.  Modular curves, Shimura curves, and Goppa codes, better than Varshamov‐Gilbert bound , 1982 .

[3]  T. Beth,et al.  On optimal quantum codes , 2003, quant-ph/0312164.

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

[5]  N. Sloane,et al.  Quantum Error Correction Via Codes Over GF , 1998 .

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

[7]  Adam D. Smith,et al.  Authentication of quantum messages , 2001, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

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

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

[10]  S. G. Vladut,et al.  Algebraic-Geometric Codes , 1991 .

[11]  Henning Stichtenoth,et al.  Algebraic function fields and codes , 1993, Universitext.

[12]  Hao Chen.  Some good quantum error-correcting codes from algebraic-Geometric codes , 2001, IEEE Trans. Inf. Theory.

[13]  J. Schwinger.  UNITARY OPERATOR BASES. , 1960, Proceedings of the National Academy of Sciences of the United States of America.

[14]  H. Stichtenoth,et al.  A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound , 1995 .

[15]  Stephen M. Barnett,et al.  Quantum information , 2005, Acta Physica Polonica A.

[16]  R. Matsumoto,et al.  Constructing Quantum Error-Correcting Codes for pm-State System from Classical Error-Correcting Codes , 1999, quant-ph/9911011.

[17]  Steane,et al.  Simple quantum error-correcting codes. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[18]  Keqin Feng,et al.  Quantum codes [[6, 2, 3]]p and [[7, 3, 3]]p (p >= 3) exist , 2002, IEEE Trans. Inf. Theory.

[19]  Eric M. Rains.  Nonbinary quantum codes , 1999, IEEE Trans. Inf. Theory.

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

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

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