Implications of ignorance for quantum-error-correction thresholds

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 the 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 two-dimensional (2D) toric code, suitable thresholds still exist by utilizing a mapping to the 2D random bond Ising model.