  A quantum stabilizer code over GF$(q)$ corresponds to a classical additive
code over GF$(q^2)$ that is self-orthogonal with respect to a symplectic inner
product. We study the decoding of quantum low-density parity-check (LDPC) codes
over binary finite fields GF$(q=2^l)$ by the sum-product algorithm, also known
as belief propagation (BP). Conventionally, a message in a nonbinary BP for
quantum codes over GF$(2^l)$ represents a probability vector over GF$(2^{2l})$,
inducing high decoding complexity. In this paper, we explore the property of
the symplectic inner product and show that scalar messages suffice for BP
decoding of nonbinary quantum codes, rather than vector messages necessary for
the conventional BP. Consequently, we propose a BP decoding algorithm for
quantum codes over GF$(2^l)$ by passing scalar messages so that it has low
computation complexity. The algorithm is specified in log domain by using
log-likelihood ratios (LLRs) of the channel statistics to have a low
implementation cost. Moreover, techniques such as message normalization or
offset can be naturally applied in this algorithm to mitigate the effects of
short cycles to improve BP performance. This is important for nonbinary quantum
codes since they may have more short cycles compared to binary quantum codes.
Several computer simulations are provided to demonstrate these advantages. The
scalar-based strategy can also be used to improve the BP decoding of classical
linear codes over GF$(2^l)$ with many short cycles.
