  Belief-propagation (BP) decoders play a vital role in modern coding theory,
but they are not suitable to decode quantum error-correcting codes because of a
unique quantum feature called error degeneracy. Inspired by an exact mapping
between BP and deep neural networks, we train neural BP decoders for quantum
low-density parity-check (LDPC) codes with a loss function tailored to error
degeneracy. Training substantially improves the performance of BP decoders for
all families of codes we tested and may solve the degeneracy problem which
plagues the decoding of quantum LDPC codes.
