  We propose a new method called decoupling representation to represent Pauli
operators as vectors over $GF(2)$, based on which we propose partially
decoupled belief propagation and fully decoupled belief propagation decoding
algorithm for quantum low density parity-check codes. These two algorithms have
the capability to deal with the correlations between the $X$ part and the $Z$
part of the vectors in symplectic representation, which are introduced by Pauli
$Y$ errors. Hence, they can not only apply to CSS codes, but also to non-CSS
codes. Under the assumption that there is no measurement error, compared with
traditional belief propagation algorithm in symplectic representation over
$GF(2)$, within the same number of iterations, the decoding accuracy of
partially decoupled belief propagation and fully decoupled belief propagation
algorithm is significantly improved in pure $Y$ noise and depolarizing noise,
which supports that decoding algorithms of quantum error correcting codes might
have better performance in decoupling representation than in symplectic
representation. The impressive performance of fully decoupled belief
propagation algorithm might promote the realization of quantum error correcting
codes in engineering.
