  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
strategies compared to established methods of decoding surface code quantum
memory (SCQM) or lattice surgery operations. In this work, we examine and
benchmark the performance of three 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. 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 works
towards systematic estimation of the cost of implementing large-scale quantum
algorithms based on transversal gates in the surface code.
