Quantum error correcting codes have a distance parameter, conveying the
minimum number of single spin errors that could cause error correction to fail.
However, the success thresholds of finite per-qubit error rate that have been
proven for the likes of the Toric code require them to work well beyond this
limit. We argue that without the assumption of being below the distance limit,
the success of error correction is not only contingent on the noise model, but
what the noise model is believed to be. Any discrepancy must adversely affect
the threshold rate, and risks invalidating existing threshold theorems. We
prove that for the 2D Toric code, suitable thresholds still exist by utilising
a mapping to the 2D random bond Ising model.Comment: 8 pages, 2 figures. Title change enforced by journa
