On Belief Propagation Decoding of Quantum Codes with Quaternary Reliability Statistics

In this paper, we investigate the use of quaternary reliability statistics for ordered statistics decoding (OSD) of quantum codes. OSD can be used to improve the performance of belief propagation (BP) decoding when it fails to correct the error syndrome. We propose an approach that leverages quaternary reliability information and the hard-decision history output by BP to perform reliability sorting for OSD. This approach improves upon previous methods that separately treat X and Z errors, by preserving the X/Z correlations during the sorting step. Our simulations show that the refined BP with scalar messages and the proposed OSD outperforms previous BP-OSD combinations. We achieve thresholds of roughly 17.5%–17.7% for toric, surface, and XZZX codes, and 15.42% for hexagonal planar color codes.