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For quantum circuits, once the gate error is below a threshold, the error probability of an entire computation can be driven exponentially small with polylog costs in time and space:

http://arxiv.org/pdf/quant-ph/9611025v2.pdf

The classical version should be entirely analogous, and was presumably proven long before the quantum case. Who proved the classical version?

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The classical theory of fault tolerance was pioneered by John von Neumann. I think this is the original reference:

von Neumann, J. (1956). "Probabilistic Logics and Synthesis of Reliable Organisms from Unreliable Components", in Automata Studies, eds. C. Shannon and J. McCarthy, Princeton University Press, pp. 43–98

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