Improving belief propagation performance for quantum error correcting codes by degeneracy-reducing transformations*

Due to its linear complexity and good error correction ability, iterative belief propagation (BP) decoder is widely used for decoding classical low-density-parity-check (LDPC) error correction codes.  However, for quantum LDPC codes, BP fails often, mainly because it cannot deal with small-weight trapping sets associated with degeneracy.  Decoding failure probability can be reduced by orders of magnitudes using, e.g., serial update schedules or otherwise breaking the symmetry, but this does not eliminate the trapping sets completely.  To address the problem, we consider BP decoding of codes specially modified to reduce the degeneracy.  The transformation amounts to a partial summation over the stabilizer group generators, adding variable nodes to account for the correlations induced.  In the spin-model language, it is a well-known star-polygon transformation used to construct marginal spin probability distributions.  We combine exact transformations with correlation-reducing approximations to optimize BP success probability and performance.