Exploiting Degeneracy in Belief Propagation Decoding of Quantum Codes

Quantum information needs to be protected by quantum error-correcting codes due to imperfect quantum devices and operations. One would like to have an efficient and high-performance decoding procedure for quantum codes. A potential candidate is Pearl's belief propagation (BP), but its performance suffers from the many short cycles inherent in quantum codes, especially \textit{{highly-degenerate}} codes (that is, codes with many low-weight stabilizers). A general impression exists that BP cannot work for topological codes, such as the surface and toric codes. In this paper, we propose a decoding algorithm for quantum codes based on quaternary BP but with additional memory effects (called MBP). This MBP is like a recursive neural network with inhibition between neurons (edges with negative weights) during recursion, which enhances the network's perception capability. Moreover, MBP exploits the degeneracy of quantum codes so that it has a better chance to find the most probable error or its degenerate errors. The decoding performance is significantly improved over the conventional BP for various quantum codes, including quantum bicycle codes, hypergraph-product codes, and surface (or toric) codes. For MBP on the surface and toric codes over depolarizing errors, we observe thresholds of 14.5\%--16\% and 14.5\%--17.5\%, respectively.