Belief propagation decoding of quantum channels by passing quantum messages

We construct a belief propagation algorithm which passes quantum messages on the factor graph and is capable of decoding the classical-quantum channel with pure state outputs. This gives explicit decoding circuits whose number of gates is quadratic in the code length. We show that the decoder can be modified to work with polar codes for the pure state channel and as part of a decoder for transmitting quantum information over the amplitude damping channel. These yield the first explicit capacity-achieving decoders for non-Pauli channels.