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The Shor code can be used to protect against an arbitrary one-qubit error. If the time interval over which quantum noise acts is short enough, can it be assumed only one-qubit errors occur, so one does not have to deal with two-qubit errors, etc.

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That might depend on your noise model. A typical noise model is independent errors on each qubit, occurring with probability $p$. In that case, as soon as there's a non-zero chance of having a single-qubit error, there's a non-zero chance of have a two-qubit error. But it could be a negligibly small probability.

Part of the point of an error correcting code is only really to reduce the probability of error. A single qubit has a probability of error $p$ while a qubit encoded in the Shor code has a probability of error of $$ 1-(1-p)^9-9p(1-p)^8\approx 36p^2-168p^3 $$ So provided $36p^2<p$, you're gaining some benefit. Encode it again, and you're doing even better. Keep on encoding, and you can get arbitrarily accurate.

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  • $\begingroup$ If $p$ is small you might argue that you can neglect the $\mathcal{O}(p^2)$ terms. $\endgroup$
    – M. Stern
    Sep 9 at 6:55
  • $\begingroup$ @M.Stern Agreed, hence my comment "could be a negligibly small probability". But without any external criterion, I don't know if I can neglect those terms. $\endgroup$
    – DaftWullie
    Sep 9 at 8:38

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