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

<p>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</p>

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2014-10-17T08:05:35.767Z
2014-10-17T08:05:35.767Z
Martin Schwarz
<p>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:</p>

<p><a href="http://arxiv.org/pdf/quant-ph/9611025v2.pdf" rel="nofollow">http://arxiv.org/pdf/quant-ph/9611025v2.pdf</a></p>

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

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2022-02-18T21:20:46.133Z
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.
Ciro Santilli OurBigBook.com
<reference-request><coding-theory>
Reference request: Classical analog of quantum threshold theorem
Geoffrey Irving
# Reference request: Classical analog of quantum threshold theorem

(Asked by: username_0 on 2014-10-16)

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?

**Tags:** reference-request, coding-theory
> **Comments:**
> **username_1**: 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.  

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## 1 answers

### Answer by username_2 on 2014-10-17. Score: 6 | Accepted answer

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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