Error correction of transversal CNOT gates for scalable surface code computation: Part 2

Recent experimental advances have made it possible to implement logical multi-qubit transversal gates on surface codes in a multitude of platforms. A transversal controlled-NOT (tCNOT) gate on two surface codes introduces correlated errors across the code blocks and thus requires modified decoding compared to established methods of decoding surface code quantum memory (SCQM) or lattice surgery operations. Here, we examine and benchmark the performance of different decoding strategies for the tCNOT for scalable, fault-tolerant quantum computation. In particular, we present a low-complexity decoder based on minimum-weight perfect matching (MWPM) that achieves the same threshold as the SCQM MWPM decoder, and discuss analytical threshold bounds for this approach. We extend our analysis with a study of tailored decoding of a transversal teleportation circuit, along with a comparison between the performance of lattice surgery and transversal operations under Pauli and erasure noise models. Our investigation builds towards systematic estimation of the cost of implementing large-scale quantum algorithms based on transversal gates in the surface code.