# If there is quantum noise, can a time interval be short enough that only one-qubit errors (eg. as in the Shor code) occur?

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.

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, you're gaining some benefit. Encode it again, and you're doing even better. Keep on encoding, and you can get arbitrarily accurate.
• If $p$ is small you might argue that you can neglect the $\mathcal{O}(p^2)$ terms. Commented Sep 9, 2021 at 6:55