A New Class of Designs Which Protect against Quantum Jumps

1. G. Alber, T. Beth, C. Charnes, A. Delgado, M. Grassl and M. Mussinger, Stabilizing distinguishable qubits against spontaneous decay by detected-jump correcting quantum codes, Physical Review Letters, Vol. 86, No. 19 (2001) pp. 4402–4405.

   Google Scholar 

2. A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin and H. Weinfurter, Elementary gates for quantum computation, Physical Review A, Vol. 52, No. 5 (1995) pp. 3457–3467.

   Google Scholar 

3. T. Beth, A class of designs protecting against quantum jumps. In (A. Blokhuis, J. W. P. Hirschfeld, D. Jungnickel and J. A. Tha, eds.), Finite Geometries, Oberwolfach: Mathematisches Forschungsinstitut, Report No. 52/2001, p. 4

4. T. Beth and M. Grassl, The quantum Hamming and hexacodes, Fortschritte der Physik, Vol. 46, No. 4–5 (1998) pp. 459–491.

   Google Scholar 

5. T. Beth, D. Jungnickel and H. Lenz, Design Theory, Encyclopaedia of Mathematics, 2nd ed., Cambridge University Press, Cambridge (1999).

   Google Scholar 

6. W. Bosma, J. J. Cannon and C. Playoust, The Magma algebra system I: the user language, Journal of Symbolic Computation, Vol. 24, No. 3–4 (1997) pp. 235–266.

   Google Scholar 

7. A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction and orthogonal geometry, Physical Review Letters, Vol. 78, No. 3 (1997a) pp. 405–408.

   Google Scholar 

8. A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), In Proceedings ISIT 97 (1997b) p. 292.

9. A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4). IEEE Transactions on Information Theory, Vol. 44, No. 4 (1998) pp. 1369–1387.

   Google Scholar 

10. A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Physical Review A, Vol. 54, No. 2 (1996) pp. 1098–1105.

    Google Scholar 

11. L. G. Chouinard II, Partitions of the 4–subset of a 13–set into disjoint projective planes, Discrete Mathematics, Vol. 45 (1983) pp. 297–300.

    Google Scholar 

12. J. I. Cirac and P. Zoller, Quantum computation with cold trapped ions, Physical Review Letters, Vol. 74, No. 20 (1995) pp. 4091–4094.

    Google Scholar 

13. C. J. Colbourn and J. H. Dinitz (eds.), The CRC Handbook of Combinatorial Designs, CRC Press (1996).

14. D. G. Cory, A. F. Fahmy and T. F. Havel, Ensemble quantum computing by nuclear resonance spectroscopy, Technical Report TR-10–96, B. C. M. P., Harvard Medical Medical School, Boston (1996).

    Google Scholar 

15. P. M. A. Dirac, The Principles of Quantum Mechanics, 4th ed., Clarendon Press, Oxford (1958).

    Google Scholar 

16. A. Ekert and C. Macchiavello, Quantum error correction for communication, Physical ReviewLetters,Vol. 77, No. 12 (1996) pp. 2585–2588.

    Google Scholar 

17. J. Fields, P. Gaborit, W. C. Huffman and V. Pless, On the classification of formally self-dual codes, In: Proceedings of the 36th Allerton Conference on Communication, Control and Computing (1998) pp. 566–575.

18. N. A. Gershenfeld and I. L. Chuang, Bulk spin-resonance quantum computation, Science, Vol. 275, No. 5298 (1997) pp. 350–356.

    Google Scholar 

19. D. Gottesman, A class of quantum error-correcting codes saturating the quantum hamming bound, Physical Review A, Vol. 54, No. 3 (1996) pp. 1862–1868.

    Google Scholar 

20. M. Grassl and T. Beth, A note on non-additive quantum codes, Technical Report quant-ph/9703016, Los Alamos National Laboratory (1997).

21. M. Grassl, T. Beth and T. Pellizzari, Codes for the quantum erasure channel, Physical Review A, Vol. 56, No. 1 (1997) pp. 33–38.

    Google Scholar 

22. M. Harada and P. R. J. Őstergård, Classification of extremal formally self-dual even codes of length 22, to appear in Graphs and Combinatorics (2002).

23. G. T. Kennedy and V. Pless, On designs and formally self-dual codes, Designs, Codes and Cryptography, (1994) pp. 43–55.

24. A. Klappenecker and M. Rötteler, Beyond stabilizer codes I: nice error bases, IEEE Transactions on Information Theory, LANL preprint quant-ph/0010082, Vol. 48, No. 8 (2002) pp. 2392–2395.

    Google Scholar 

25. E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Physical Review A, Vol. 55, No. 2 (1997) pp. 900–911.

    Google Scholar 

26. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam (1977).

    Google Scholar 

27. T. Pellizzari, T. Beth, M. Grassl and J. Müller-Quade, Stabilization of quantum states in quantum optical systems, Physical Review A, Vol. 54, No. 4 (1996) pp. 2698–2702.

    Google Scholar 

28. T. Pellizzari, S. A. Gardiner, J. I. Cirac and P. Zoller, Decoherence, continuous observation, and quantum computing: a cavity QED model, Physical Review Letters, Vol. 75, No. 21 (1995) pp. 3788–3791.

    Google Scholar 

29. E. M. Rains and N. J. A. Sloane, Self-dual codes. In (V. P. Pless and W. C. Huffman eds.), Handbook of Coding Theory, North-Holland (1998).

30. P. W. Shor, Algorithms for quantum computation: discrete logarithm and factoring. In Proceedings of the 35th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press (1994) pp. 124–134.

31. P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Physical Review A, Vol. 52, No. 4 (1995) pp. R2493–R2496.

    Google Scholar 

32. J. Simonis, The [18, 9, 6] code is unique, Discrete Mathematics, Vol. 106/107 (1992) pp. 439–448.

    Google Scholar 

33. A. Steane, Multiple particle interference and quantum error correction, Proceedings of the Royal Society London Series A, Vol. 452 (1996a) pp. 2551–2577.

    Google Scholar 

34. A. M. Steane, Error correcting codes in quantum theory, Physical Review Letters, Vol. 77, No. 5 (1996b) pp. 793–797.

    Google Scholar 

35. W. K. Wootters and W. H. Zurek, A single quantum cannot be cloned, Nature, Vol. 299, No. 5886 (1982) pp. 802–803.

    Google Scholar 

Download references