  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.
