  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.
