Improved Belief Propagation Decoding Algorithms for Surface Codes

Quantum error correction is crucial for universal fault-tolerant quantum computing. Highly accurate and low-time-complexity decoding algorithms play an indispensable role in ensuring quantum error correction works effectively. Among existing decoding algorithms, belief propagation (BP) is notable for its nearly linear time complexity and general applicability to stabilizer codes. However, BP's decoding accuracy without postprocessing is unsatisfactory in most situations. This article focuses on improving the decoding accuracy of BP over GF(4) for surface codes. Inspired by machine learning optimization techniques, we first propose Momentum-BP and AdaGrad-BP to reduce oscillations in message updating, breaking the trapping sets of surface codes. We further propose exponential weighted average initialization belief propagation (EWAInit-BP), which adaptively updates initial probabilities and provides a one to three orders of magnitude improvement over traditional BP for planar surface code, toric code, and $XZZX$ surface code without any postprocessing method, showing high decoding accuracy even under parallel scheduling. The theoretical $O(1)$ time complexity under parallel implementation and high accuracy of EWAInit-BP make it a promising candidate for high-precision real-time decoders.