  We address the problem of decoding sparse quantum error correction codes. For
Pauli channels, this task can be accomplished by a version of the belief
propagation algorithm used for decoding sparse classical codes. Quantum codes
pose two new challenges however. Firstly, their Tanner graph unavoidably
contain small loops which typically undermines the performance of belief
propagation. Secondly, sparse quantum codes are by definition highly
degenerate. The standard belief propagation algorithm does not exploit this
feature, but rather it is impaired by it. We propose heuristic methods to
improve belief propagation decoding, specifically targeted at these two
problems. While our results exhibit a clear improvement due to the proposed
heuristic methods, they also indicate that the main source of errors in the
quantum coding scheme remains in the decoding.
