Implications of ignorance for quantum-error-correction thresholds

Abstract

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.

Quantum error correction for state transfer in noisy spin chains

“…We are given heart by the surface code [39] -this is a CSS code and, although its distance is short, it is the case that in the presence of local noise, it is highly unlikely that those short error strings that cause failure of the code arise. Indeed, the surface code has an error correcting threshold consisting of a finite per-qubit error rate, exactly as we desire [40,41]. However, more work would be required -the errors in the present model are not independent.…”

A family of stabilizer codes for $D({{\mathbb{Z}}_{2}})$ anyons and Majorana modes

“…The potential that  D ( ) 2 codes have for quantum computation is well recognized, though most focus is on the square lattice planar variant of the surface code. However, it is known that this does not provide the best protection against every error model [24][25][26]. The matching codes therefore provide a new family of codes to consider for the optimization of error correction against physical error models.…”