Decoding across transversal logical Clifford gates: the S gate for surface code computation

Some quantum computing platforms, such as neutral atoms, allow for the connectivity required to run fold-transversal Clifford gates on the surface code (SC). This enables fast computation with O(1) quantum error correction (QEC) rounds between logical gates, but puts higher demands on decoding. Minimum Weight Perfect Matching (MWPM) is an efficient algorithm that achieves a high threshold for SC idling, but can fail to decode transversal Clifford gates due to the presence of hyperedges in the decoding graph. For the transversal CNOT, alternative algorithms have been proposed [Sahay, K. et al. arXiv2408.01393, Wan, K.H. et al. arXiv2407.20976]. We present a low-complexity decoder for the fold-transversal S gate, which MWPM poorly decodes. Our algorithm achieves exponential error suppression below p ≈ 0.2% for circuit-level noise with a single QEC round between S gates (same as the inefficient BPOSD decoder). In principle, our approach can decode any transversal gate that maps Z-stabilizer generators to themselves. We discuss integrating this method into a decoder that decodes across all transversal Clifford gates of the SC.