Generalized Belief Propagation Algorithms for Decoding of Surface Codes

Quantum 7, 1037 (2023).

https://doi.org/10.22331/q-2023-06-07-1037

Belief propagation (BP) is well-known as a low complexity decoding algorithm with a strong performance for important classes of quantum error correcting codes, e.g. notably for the quantum low-density parity check (LDPC) code class of random expander codes. However, it is also well-known that the performance of BP breaks down when facing topological codes such as the surface code, where naive BP fails entirely to reach a below-threshold regime, i.e. the regime where error correction becomes useful. Previous works have shown, that this can be remedied by resorting to post-processing decoders outside the framework of BP. In this work, we present a generalized belief propagation method with an outer re-initialization loop that successfully decodes surface codes, i.e. opposed to naive BP it recovers the sub-threshold regime known from decoders tailored to the surface code and from statistical-mechanical mappings. We report a threshold of $textit{17%}$ under independent bit-and phase-flip data noise (to be compared to the ideal threshold of $textit{20.6%}$) and a threshold value of $textit{14%}$ under depolarizing data noise (compared to the ideal threshold of $textit{18.9%}$), which are on par with thresholds achieved by non-BP post-processing methods.