Reliable Message Passing in Topological Quantum Codes

This work takes initial steps towards more rigorous understanding of message passing and reliable Belief Propagation (BP) decoding in quantum error correction. BP is asymptotically correct on sparse, locally tree-like graphs, explaining its success for classical LDPC codes. In contrast, topological quantum codes such as surface codes must obey a global commutativity constraint that introduces dense short loops and high treewidth. Yet this same constraint also induces degeneracy, where many errors belong to equivalent homological classes and are equally correctable. In this work, we turn this structural redundancy from an obstacle into a resource by introducing a graph-dilution method that exploits stabilizer symmetries and percolation insights into error geometry to progressively sparsify the decoding graph of surface codes. This dilution coarse-grains the original Tanner graph into a locally tree-like structure, enabling standard quaternary BP with near-linear complexity to converge reliably. Without heuristic parameter tuning or post-processing, BP on the diluted graph achieves a 16% threshold for surface codes under depolarizing noise, surpassing MWPM and BP-OSD in both threshold and finite-length performance.