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Belief propagation is a powerful tool in statistical physics, machinelearning, and modern coding theory. As a decoding method, it is ubiquitous inclassical error correction and has also been applied to stabilizer-basedquantum error correction. The algorithm works by passing messages between nodesof the factor graph associated with the code and enables efficient decoding, insome cases even up to the Shannon capacity of the channel. Here we construct abelief propagation algorithm which passes quantum messages on the factor graphand is capable of decoding the classical-quantum channel with pure stateoutputs. This gives explicit decoding circuits whose number of gates isquadratic in the blocklength of the code. We also show that this decoder can bemodified to work with polar codes for the pure state channel and as part of apolar decoder for transmitting quantum information over the amplitude dampingchannel. These represent the first explicit capacity-achieving decoders fornon-Pauli channels.

Belief propagation decoding of quantum channels by passing quantum messages