Belief Propagation Decoding on a Sparsified Graph Ensemble of the Surface Code
This work exploits the product structure of surface codes to improve their belief propagation (BP) decoding performance. To promote BP convergence, we sparsify the original surface code graph with respect to its topological product structure to break its dominant short cycles. Analytical justification of the proposed sparsification methods is provided through analysis and minimum-weight perfect matching (MWPM) on sparsified graphs. The results demonstrate that the increase in logical error is well-controlled in both worst-case and average-case scenarios. Practically, we show that BP on a single sparsified graph significantly outperforms BP on the original graph in terms of convergence and total error rate. To further address the performance loss relative to MWPM caused by sparsification, we construct an ensemble of sparsified decoding graphs. Numerical results show that running standard BP on the ensemble, without any post-processing or auxiliary subroutines, achieves a threshold of 9. 0 \% on surface codes under Pauli- X noise, closely matching the BP-OSD threshold of approximately 9. 1 \%.
