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:


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

  • $\begingroup$ I wonder what more is there to say for the classical case besides: "the input of the component is considered a logical 0 if the voltage is below 1.5V, and and a logical 0 output is 1.0V. Because 1.0V < 1.5V, everything works out fine". Analogous for 1 but with voltages inverted. $\endgroup$ Commented Feb 18, 2022 at 21:20

1 Answer 1


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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