Message passing in fault-tolerant quantum error correction

Inspired by Knill's scheme for message passing error detection, here we develop a scheme for message passing error correction for the nine-qubit Bacon-Shor code. We show that for two levels of concatenated error correction, where classical information obtained at the first level is used to help interpret the syndrome at the second level, our scheme will correct all cases with four physical errors. This results in a reduction of the logical failure rate relative to conventional error correction by a factor proportional to the reciprocal of the physical error rate.