  Quantum stabilizer codes constructed from sparse matrices have good
performance and can be efficiently decoded by belief propagation (BP). A
conventional BP decoding algorithm treats binary stabilizer codes as additive
codes over GF(4). This algorithm has a relatively complex process of handling
check-node messages, which incurs higher decoding complexity. Moreover, BP
decoding of a stabilizer code usually suffers a performance loss due to the
many short cycles in the underlying Tanner graph. In this paper, we propose a
refined BP decoding algorithm for quantum codes with complexity roughly the
same as binary BP. For a given error syndrome, this algorithm decodes to the
same output as the conventional quaternary BP but the passed node-to-node
messages are single-valued, unlike the quaternary BP, where multivalued
node-to-node messages are required. Furthermore, the techniques of message
strength normalization can naturally be applied to these single-valued messages
to improve the performance. Another observation is that the message-update
schedule affects the performance of BP decoding against short cycles. We show
that running BP with message strength normalization according to a serial
schedule (or other schedules) may significantly improve the decoding
performance and error floor in computer simulation.
