  A stabilizer code over GF($q=2^l$) corresponds to a classical additive code
over GF($q^2 = 2^{2l}$) that is self-orthogonal with respect to a binary
symplectic inner product. We study the decoding of quantum low-density
parity-check (LDPC) codes over finite fields GF($q=2^l$) for $l\ge 1$ by the
sum-product algorithm (SPA), also known as belief propagation (BP). Previously,
all the BP decoding of quantum codes are studied in linear domain. In this
paper, we propose a BP decoding algorithm for quantum codes over GF($2^l$) in
log domain by passing scalar messages derived from log-likelihood ratios (LLRs)
of the channel statistics. Techniques such as message normalization or offset
for improving BP performance can be naturally applied in this algorithm.
Moreover, the implementation cost of this LLR-BP is relatively small compared
to the linear-domain BP, which is of the same case as in the classical coding
research. Several computer simulations are provided to demonstrate these
advantages.
