Generalized Belief Propagation Algorithms for Decoding of Surface Codes

TL;DR

This paper introduces a generalized belief propagation algorithm with an outer re-initialization loop that effectively decodes surface codes, achieving error correction thresholds comparable to specialized non-BP methods.

Contribution

The work presents a novel generalized belief propagation method with re-initialization that successfully decodes surface codes, overcoming limitations of naive BP.

Findings

Achieved a 17% threshold under bit-and phase-flip noise.

Achieved a 14% threshold under depolarizing noise.

Thresholds are comparable to non-BP post-processing decoders.

Abstract

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 17% under independent bit-and phase-flip data noise (to be compared to the ideal threshold of 20.6%) and a threshold value of 14%$under depolarizing data noise (compared to the ideal threshold of 18.9%), which are on par with thresholds achieved by non-BP post-processing methods.