It is well established that there exists a noise threshold for quantum computation, such that below this threshold, the computation can be encoded in such a way that it yields the correct result with bounded probability (with at most polynomial computational overhead). This threshold depends on the encoding used and the exact nature of the noise, and it is the case that results from simulation often give thresholds much higher than what can be proved for adversarial noise models.

So my question is simply what is the highest lower bound that has been
proved for independent stochastic noise?

The noise model I am referring to is the one dealt with in quant-ph/0504218, where Aliferis, Gottesman and Preskill prove a lower bound $2.73 \times 10^{-5}$. Note, however, I do not care which type of encoding is used, and it need not be restricted to the code considered in that paper. The highest I'm aware of is $1.94 \times 10^{-4}$ due to Aliferis and Cross (quant-ph/0610063). Has this value been improved upon since then?

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