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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?
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
