-
1
-
-
33746738958
-
-
PLRAAN
-
P. W. Shor, Phys. Rev. A 52, 2493 (1995).PLRAAN
-
(1995)
Phys. Rev. A
, vol.52
, pp. 2493
-
-
Shor, P.W.1
-
2
-
-
4243789313
-
-
PRLTAO
-
C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, 722 (1996).PRLTAO
-
(1996)
Phys. Rev. Lett.
, vol.76
, pp. 722
-
-
Bennett, C.H.1
Brassard, G.2
Popescu, S.3
Schumacher, B.4
Smolin, J.A.5
Wootters, W.K.6
-
3
-
-
4243216277
-
-
PLRAAN
-
C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824 (1996).PLRAAN
-
(1996)
Phys. Rev. A
, vol.54
, pp. 3824
-
-
Bennett, C.H.1
DiVincenzo, D.P.2
Smolin, J.A.3
Wootters, W.K.4
-
13
-
-
0000132092
-
-
PRLTAO
-
R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Phys. Rev. Lett. 77, 198 (1996); PRLTAO
-
(1996)
Phys. Rev. Lett.
, vol.77
, pp. 198
-
-
Laflamme, R.1
Miquel, C.2
Paz, J.P.3
Zurek, W.H.4
-
16
-
-
0001060859
-
-
PRLTAO
-
A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Phys. Rev. Lett. 78, 405 (1996).PRLTAO
-
(1996)
Phys. Rev. Lett.
, vol.78
, pp. 405
-
-
Calderbank, A.R.1
Rains, E.M.2
Shor, P.W.3
Sloane, N.J.A.4
-
22
-
-
4243536108
-
-
PLRAAN
-
R. Cleve, Phys. Rev. A 55, 4054 (1997).PLRAAN
-
(1997)
Phys. Rev. A
, vol.55
, pp. 4054
-
-
Cleve, R.1
-
23
-
-
85037195680
-
-
IEEE Trans. Inf. Theory
-
E. M. Rains, IEEE Trans. Inf. Theory (to be published);
-
-
-
Rains, E.M.1
-
25
-
-
85037192891
-
-
Quantum hash coding is a protocol for which a vanishingly small fraction of errors are not corrected. The distance (Formula presented) for a typical hash code can be obtained by the following counting procedure: There are (Formula presented) likely error patterns to be assigned to (Formula presented) possible hash strings. It is highly likley that there will be cases where two error patterns are at random assigned to the same hash. A smaller set of error patterns can be assigned uniquely to hashes with a low probability of collision, provided that this number is smaller than (Formula presented) (coming from the usual birthday-paradox reasoning). This fixes the distance between such errors according to (Formula presented). Thus the distance scales as the block size, being approximately (Formula presented)
-
Quantum hash coding is a protocol for which a vanishingly small fraction of errors are not corrected. The distance (Formula presented) for a typical hash code can be obtained by the following counting procedure: There are (Formula presented) likely error patterns to be assigned to (Formula presented) possible hash strings. It is highly likley that there will be cases where two error patterns are at random assigned to the same hash. A smaller set of error patterns can be assigned uniquely to hashes with a low probability of collision, provided that this number is smaller than (Formula presented) (coming from the usual birthday-paradox reasoning). This fixes the distance between such errors according to (Formula presented). Thus the distance scales as the block size, being approximately (Formula presented).
-
-
-
-
27
-
-
84856043672
-
-
Bell Syst. Tech. J.C. E. Shannon27, 623 (1948).
-
(1948)
, vol.27
, pp. 623
-
-
Shannon, C.E.1
-
29
-
-
85037190285
-
-
Of course, even more exotic codes exist, ones that do not conform to the additive code classification at all. The only nonadditive codes of which we are aware are in
-
Of course, even more exotic codes exist, ones that do not conform to the additive code classification at all. The only nonadditive codes of which we are aware are in 42425. We cannot rule out the possibility that a nonadditive code achieves the best possible capacity for depolarizing channels. Only the code in 25 is directly relevant the depolarizing channel discussed in most of this paper.
-
-
-
-
30
-
-
4243489981
-
-
PLRAAN
-
D. W. Leung, M. A. Nielsen, I. L. Chuang, and Y. Yamamoto, Phys. Rev. A 56, 2567 (1997).PLRAAN
-
(1997)
Phys. Rev. A
, vol.56
, pp. 2567
-
-
Leung, D.W.1
Nielsen, M.A.2
Chuang, I.L.3
Yamamoto, Y.4
-
31
-
-
6144225745
-
-
PRLTAO
-
E. M. Rains, R. H. Hardin, P. W. Shor, and N. J. A. Sloane, Phys. Rev. Lett. 79, 953 (1997).PRLTAO
-
(1997)
Phys. Rev. Lett.
, vol.79
, pp. 953
-
-
Rains, E.M.1
Hardin, R.H.2
Shor, P.W.3
Sloane, N.J.A.4
-
32
-
-
85037217654
-
-
Proceedings of the Symposium on the Foundations of Computer Science (IEEE Press, Los Alamitos, CA, 1996)
-
P. W. Shor, in Proceedings of the Symposium on the Foundations of Computer Science (IEEE Press, Los Alamitos, CA, 1996);
-
-
-
Shor, P.W.1
-
39
-
-
0001521178
-
-
PLRAAN
-
S. Lloyd, Phys. Rev. A 55, 1613 (1997).PLRAAN
-
(1997)
Phys. Rev. A
, vol.55
, pp. 1613
-
-
Lloyd, S.1
-
41
-
-
4243059985
-
-
PRLTAO
-
C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).PRLTAO
-
(1993)
Phys. Rev. Lett.
, vol.70
, pp. 1895
-
-
Bennett, C.H.1
Brassard, G.2
Crépeau, C.3
Jozsa, R.4
Peres, A.5
Wootters, W.K.6
-
42
-
-
51349163744
-
-
JMOPEW
-
R. Jozsa, J. Mod. Opt. 41, 2315 (1994).JMOPEW
-
(1994)
J. Mod. Opt.
, vol.41
, pp. 2315
-
-
Jozsa, R.1
-
44
-
-
85037249056
-
-
We did not search all possible cat-in-cat codes up to 25 qubits, but the next best code we identified was a (Formula presented) inside a (Formula presented) cat code which had a threshold at (Formula presented). The (Formula presented) inside a (Formula presented) code was much worse, having a threshold at (Formula presented), worse even than the regular hashing result for which (Formula presented) and the threshold is at (Formula presented)
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We did not search all possible cat-in-cat codes up to 25 qubits, but the next best code we identified was a (Formula presented) inside a (Formula presented) cat code which had a threshold at (Formula presented). The (Formula presented) inside a (Formula presented) code was much worse, having a threshold at (Formula presented), worse even than the regular hashing result for which (Formula presented) and the threshold is at (Formula presented).
-
-
-
-
45
-
-
85037196224
-
-
D. Bruss, D. P. DiVincenzo, A. Ekert, C. Fuchs, C. Macchiavello, and A. Smolin, quant-ph/9705038.
-
-
-
Bruss, D.1
DiVincenzo, D.P.2
Ekert, A.3
Fuchs, C.4
Macchiavello, C.5
Smolin, A.6
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