Quantum error correction via codes over GF (4)

IEEE Transactions on Information Theory

Volumn 44, Issue 4, 1998, Pages 1369-1387

  Robert Calderbank, A ^a   Rains, Eric M ^a   Shor, P W ^a   Sloane, Neil J A ^a  

^a AT AND T LABS RESEARCH   (United States)

Author keywords

Codes, additive; Codes, quantum; Codes, quaternary; Codes, self orthogonal, geometry, orthogonal; Geometry, symplectic; Group, clifford

Indexed keywords

CODES (SYMBOLS); ERROR CORRECTION; SIGNAL ENCODING;

QUANTUM ERROR CORRECTING CODES;

INFORMATION THEORY;

EID: 0032121784     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/18.681315     Document Type: Article

References (76)

1. M. Aschbacher, Finite Group Theory. Cambridge, U.K.: Cambridge Univ. Press, 1986.
2. D. Aharonov and M. Ben-Or, "Fault-tolerant quantum computation with constant error," in Proc. 29th Anna. ACM Symp. Theory of Computing. ACM Press, 1997, pp. 176-188; also LANL e-print quant-ph/9611025.
3. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, "Teleporting an unknown quantum state via dual classical and EPR channels," Phys. Rev. Lett., vol. 70, pp. 1895-1898, 1993.
4. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, "Purification of noisy entanglement and faithful teleportation via noisy channels," Phys. Rev. Lett., vol. 76, pp. 722-725, 1996; also LANL e-print quant-ph/9511027.
5. C. H. Bennett, D. DiVincenzo, J. A. Smolin, and W. K. Wootters, "Mixed state entanglement and quantum error correction," Phys. Rev. A, vol. 54, pp. 3824-3851, 1996; also LANL e-print quant-ph/9604024.
6. B. Bolt, T. G. Room, and G. E. Wall, "On Clifford collineation, transform and similarity groups I," J. Australian Math. Soc., vol. 2, pp. 60-79, 1961.
7. _, "On Clifford collineation, transform and similarity groups II," J. Australian Math. Soc., vol. 2, pp. 80-96, 1961.
8. W. Bosma and J. Cannon, Handbook of Magma Functions, Sydney, May 22, 1995.
9. W. Bosma, J. J Cannon, and G. Mathews, "Programming with algebraic structures: Design of the Magma language," in Proc. 1994 Int. Symp. Symbolic and Algebraic Computation, M. Giesbrecht, Ed. (Oxford, U.K., July 20-22, 1994). ACM, 1994, pp. 52-57.
10. W. Bosma, J. Cannon, and C. Playoust, "The Magma algebra system I: The user language," J. Symb. Comp., vol. 24, pp. 235-265, 1997.
11. A. E. Brouwer, "Tables of bounds on linear codes," in Handbook of Coding Theory, V. Pless et al. Eds., 1998, to be published.
12. A. R. Calderbank, P. J. Cameron, W. M. Kantor, and J. J. Seidel, "Z4 Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets," Proc. London Math. Soc., vol. 75, pp. 436-480, 1997.
13. A. R. Calderbank, R. H. Hardin, E. M. Rains, P. W. Shor, and N. J. A. Sloane, "A group-theoretic framework for the construction of packings in Grassmannian spaces," J. Alg. Comb., 1997, submitted for publication.
14. A. R. Calderbank and W. M. Kantor, "The geometry of two-weight codes," Bull. London Math. Soc., vol. 118, pp. 97-122, 1986.
15. A. R. Calderbank, W.-C. W. Li, and B. Poonen, "A 2-adic approach to the analysis of cyclic codes," IEEE Trans. Inform. Theory, vol. 43, pp. 977-986, May 1997.
16. A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, "Quantum error correction and orthogonal geometry," Phys. Rev. Lett., vol. 78, pp. 40509, 1997; also LANL e-print quant-ph/9605005.
17. A. R. Calderbank and P. W. Shor, "Good quantum error-correcting codes exist," Phys. Rev. A, vol. 54, pp. 1098-1105, 1996; also LANL e-print quant-ph/9512032.
18. B. W. Char et al.. Maple V Library Reference Manual. New York: Springer-Verlag, 1991.
19. C. Chevalley, The Construction and Study of Certain Important Algebras, Math. Soc. Japan, 1955. Reprinted with corrections in C. Chevalley, Collected Works, vol. 2. New York: Springer-Verlag, 1997.
20. R. Cleve, "Quantum stabilizer codes and classical linear codes," Phys. Rev. A, vol. 55, pp. 4054059, 1997; also LANL e-print quantph/9612048.
21. R. Cleve and D. Gottesman, "Efficient computations of encodings for quantum error correction," Phys. Rev. A., vol. 56, pp. 76-82, 1997; also LANL e-print quant-ph/9607030.
22. J. H. Conway, R. H. Hardin, and N. J. A. Sloane, "Packing lines, planes, etc.: Packings in Grassmannian space," Exper. Math., vol. 5, pp. 139-159, 1996.
23. J. H. Conway, S. P. Norton, R. A. Parker, and R. A. Wilson, ATLAS of Finite Groups. Oxford, U.K.: Oxford Univ. Press, 1985.
24. J. H. Conway, V. Pless, and N. J. A. Sloane, "Self-dual codes over G F (3) and GF(4) of length not exceeding 16," IEEE Trans. Inform. Theory, vol. IT-25, pp. 312-322, 1979.
25. J. H. Conway and N. J. A. Sloane, "A new upper bound on the minimal distance of self-dual codes," IEEE Trans. Inform. Theory, vol. 36, pp. 1319-1333, 1990.
26. _, Sphere Packings, Lattices and Groups, 2nd ed. New York: Springer-Verlag, 1993.
27. CPLEX Manual, CPLEX Organization Inc., Incline Village, NV, 1991.
28. P. Delsarte, "Bounds for unrestricted codes, by linear programming," Philips Res. Rep., vol. 27, pp. 272-289, 1972.
29. D. Dieks, "Communication by EPR devices," Phys. Lett. A, vol. 92, p. 271, 1982.
30. D. P. DiVincenzo and P. W. Shor, "Fault-tolerant error correction with efficient quantum codes," Phys. Rev. Lett., vol. 77, pp. 3260-3263, 1996; also LANL e-print quant-ph/9605031.
31. D. P. DiVincenzo, P. W. Shor, and J. A. Smolin, "Quantum channel capacity of very noisy channels," Phys. Rev. A, vol. 57, pp. 830-839, 1998, also LANL e-print quant-ph/9706061.
32. A. S. Drud, "CONOPT-A large scale GRG code," ORSA J. Comput., vol. 6, pp. 207-218, 1994.
33. W. Duke, "On codes and Siegel modular forms," Int.. Math. Res. Notices, vol. 5, pp. 125-136, 1993.
34. A. Ekert and C. Macchiavello, "Error correction in quantum communication," Phys. Rev. Lett., vol. 77, pp. 2585-2588, 1996; also LANL e-print quant-ph/9602022.
35. R. Fourer, D. M. Gay, and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming. San Francisco, CA: Scientific, 1993.
36. D. Gottesman, "A class of quantum error-correcting codes saturating the quantum Hamming bound," Phys. Rev. A, vol. 54, pp. 1862-1868, 1996; also LANL e-print quant-ph/9604038.
37. _, "A theory of fault-tolerant quantum computation," Phys. Rev. A, vol. 57, pp. 127-137, 1998, also LANL e-print quant-ph/9702029.
38. _, "Pasting quantum codes," LANL e-print quant-ph/9607027.
39. T. A. Gulliver and V. K. Bhargava, "Some best rate 1/p and rate (p -1)/P systematic quasi-cyclic codes over GF(3) and GF(4)," IEEE Trans. Inform. Theory, vol. 38, pp. 1369-1374, 1992.
40. G. Höhn, "Self-dual codes over the Kleinian four group," preprint, Aug. 16 , 1996.
41. B. Huppert, Endliche Gruppen. Berlin, Germany: Springer-Verlag, 1967.
42. N. Jacobson, Basic Algebra II. San Francisco, CA: Freeman 1980.
43. L. S. Kazarin, "On the Sidelnikov group" (in Russian), preprint, 1997.
44. A. Kitaev, personal communication, 1997.
45. E. Knill, "Non-binary unitary error bases and quantum codes," LANL e-print quant-ph/9608037.
46. _, "Group representations, error bases and quantum codes," LANL e-print quant-ph/9608048.
47. E. Knill and R. Laflamme, "A theory of quantum error-correcting codes," Phys. Rev. A. vol. 55, pp. 900-911, 1997; LANL e-print quantph/9604034.
48. E. Knill, R. Laflamme, and W. Zurek, "Threshold accuracy for quantum computation," LANL e-print quant-ph/9610011. See also, E. Knill, R. Laflamme, and W. Zurek, "Resilient quantum computation," Science, vol. 279, pp. 342-345, 1998.
49. F. R. Kschischang and S. Pasupathy, "Some ternary and quaternary codes and associated sphere packings," IEEE Trans. Inform. Theory, vol. 38, pp. 227-246, 1992.
50. R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, "Perfect quantum error correction code," Phys. Rev. Lett., vol. 77, pp. 198-201, 1996; also LANL e-print quant-ph/9602019.
51. C. W. H. Lam and V. Pless, "There is no (24,12,10) self-dual quaternary code," IEEE Trans. Inform. Theory, vol. 36, pp. 1153-1156, 1990.
52. J. H. van Lint, Introduction to Coding Theory. New York: Springer-Verlag, 1982.
53. F. J. MacWilliams, A. M. Odlyzko, N. J. A. Sloane, and H. N. Ward, "Self-dual codes over G.F(4)," J. Comb. Theory, Ser. A, vol. 25, pp. 288-318, 1978.
54. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977. [55] C. L. Mallows and N. J. A. Sloane, "An upper bound for self-dual codes," Inform. Contr., vol. 22, pp. 188-200, 1973.
55. M. Oura, "The dimension formula for the ring of code polynomials in genus 4," Osaka J. Math., vol. 34, pp. 53-72, 1997.
56. E. M. Rains, "Shadow bounds for self-dual codes," IEEE Trans. Inform. Theory, vol. 44, pp. 134-139, Jan. 1998.
57. _, "Quantum shadow enumerators," LANL e-print quant-ph/9611001.
58. _, "Quantum weight enumerators," this issue, pp. 1388-1394, also LANL e-print quant-ph/9612015.
59. _, "Nonbinary quantum codes," LANL e-print quant-ph/9703048.
60. _, "Quantum codes of minimum distance two," LANL e-print quant-ph/9704043.
61. E. M. Rains, R. H. Hardin, P. W. Shor, and N. J. A. Sloane, "A nonadditive quantum code," Phys. Rev. Lett., vol. 79, pp. 953-954, 1997; also LANL e-print quant-ph/9703002.
62. B. Runge, "On Siegel modular forms I," J. Reine Angew. Math., vol. 436, pp. 57-85, 1993.
63. _, "On Siegel modular forms II," Nagoya Math. J., vol. 138, pp. 179-197, 1995.
64. P. W. Shor, "Scheme for reducing decoherence in quantum memory," Phys. Rev. A, vol. 52, p. 2493, 1995.
65. _, "Fault-tolerant quantum computation," in P roc. 37th Symp.Foundations of Computer Science. IEEE Computer Soc. Press, 1996, pp. 56-65; also LANL e-print quant-ph/9605011.
66. P. W. Shor and R. Laflamme, "Quantum analog of the MacWilliams identities in classical coding theory," Phys. Rev. Lett., vol. 78, pp. 1600-1602, 1997; also LANL e-print quant-ph/9610040.
67. P. W. Shor and N. J. A. Sloane, "A family of optimal packings in Grassmannian manifolds," J. Alg. Comb., vol. 7, pp. 157-163, 1998.
68. V. M. Sidelnikov, "On a finite group of matrices and codes on the Euclidean sphere" Probt. Pered. Inform., vol. 33, pp. 35-54, 1997 (in Russian).
69. _, "On a finite group of matrices generating orbit codes on the Euclidean sphere," in Proc. 1997 IEEE Int. Symp. Information Theory (Ulm, Germany, 1997). New York: IEEE Press, 1997, p. 436.
70. N. J. A. Sloane, "Self-dual codes and lattices," in Relations Between Combinatorics and Other Parts of Mathematics, Proc. Symp. Pure Math., vol. 34. Providence, RI: Amer. Math. Soc., 1979, pp. 273-308.
71. A. M. Steane, "Multiple particle interference and quantum error correction," Proc. Roy. Soc. London A, vol. 452, pp. 2551-2577, 1996; also LANL e-print quant-ph/9601029.
72. _, "Simple quantum error correcting codes," Phys. Rev. Lett., vol. 77, pp. 793-797, 1996; also LANL e-print quant-ph/9605021.
73. _, "Quantum Reed-Muller codes," LANL e-print quantph/9608026.
74. _, "Space, time, parallelism and noise requirements for reliable quantum computing," LANL e-print quant-ph/9708021.
75. G. E. Wall, "On Clifford collineation, transform and similarity groups IV," Nagoya Math. J., vol. 21, pp. 199-222, 1962.
76. W. K. Wootters and W. H. Zurek, "A single quantum cannot be cloned," Nature, vol. 299, p. 802, 1982.