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A 452, 2551 (1996). [43] A. Ekert and C. Macchiavello, Phys. Rev. Lett. 77, 2585 (1996). [44] R. Laflamme, C. P. H. Zurek, Phys. Rev. Lett. 77, 198 (1996). [45] D. Gottesman, Phys. Rev. A 54, 1862 (1996). R. M. W. A. Sloane, Phys. Rev. Lett. 78, 405 (1997). [47] E. Knill and R. Laflamme, e-print quant-ph/9608012 (1996). W. P. W. Shor, Phys. Rev. Lett. 77, 3260 (1996). Solovay, “Lie groups and quantum circuits”, preprint 1999.

If δ(n) = 1 − A exp(−αn), where A and α are positive constants, then the randomised algorithm cannot technically be regarded as efficient any more regardless of how weak the coupling to the environment may be. Unfortunately, the computer-environment interaction leads to just such an unwelcome exponential increase of the error rate with the input size. To see this consider a register of size n and assume that each qubit decoheres separately, | x | M = | xn−1 . . x1 x0 | m . . | m | m → | xn−1 .

Ekert, Phys. Rev. Lett. 67, 661 (1991). [38] W. H. Zurek, Phys. 36 (1991). [39] A. Berthiaume, D. Deutsch and R. Jozsa, Proceedings of the Workshop on the Physics and Computation—PhysComp ’94, IEEE Computer Society Press, Dallas, Texas (1994); A. Barenco, A. Berthiaume, D. Deutsch, A. Ekert, R. Jozsa and C. Macchiavello, SIAM J. Comput. 26, 1541 (1997). [40] P. W. Shor, Phys. Rev. A 52, R2493 (1995). [41] R. Calderbank and P. W. Shor, Phys. Rev. A 54, 1098 (1996). [42] A. Steane, Phys. Rev. Lett.

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