By Artur Ekert, Patrick Hayden and Hitoshi Inamori.
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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 .
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