  In this work, we propose a fully differentiable iterative decoder for quantum
low-density parity-check (LDPC) codes. The proposed algorithm is composed of
classical belief propagation (BP) decoding stages and intermediate graph neural
network (GNN) layers. Both component decoders are defined over the same sparse
decoding graph enabling a seamless integration and scalability to large codes.
The core idea is to use the GNN component between consecutive BP runs, so that
the knowledge from the previous BP run, if stuck in a local minima caused by
trapping sets or short cycles in the decoding graph, can be leveraged to better
initialize the next BP run. By doing so, the proposed decoder can learn to
compensate for sub-optimal BP decoding graphs that result from the design
constraints of quantum LDPC codes. Since the entire decoder remains
differentiable, gradient descent-based training is possible. We compare the
error rate performance of the proposed decoder against various post-processing
methods such as random perturbation, enhanced feedback, augmentation, and
ordered-statistics decoding (OSD) and show that a carefully designed training
process lowers the error-floor significantly. As a result, our proposed decoder
outperforms the former three methods using significantly fewer post-processing
attempts. The source code of our experiments is available online.
