  Codes based on sparse matrices have good performance and can be efficiently
decoded by belief-propagation (BP). Decoding binary stabilizer codes needs a
quaternary BP for (additive) codes over GF(4), which has a higher check-node
complexity compared to a binary BP for codes over GF(2). Moreover, BP decoding
of stabilizer codes suffers a performance loss from the short cycles in the
underlying Tanner graph. In this paper, we propose a refined BP algorithm for
decoding quantum codes by passing scalar messages. For a given error syndrome,
this algorithm decodes to the same output as the conventional quaternary BP but
with a check-node complexity the same as binary BP. As every message is a
scalar, the message normalization can be naturally applied to improve the
performance. Another observation is that the message-update schedule affects
the BP decoding performance against short cycles. We show that running BP with
message normalization according to a serial schedule (or other schedules) may
significantly improve the decoding performance and error-floor in computer
simulation.
