Logical operations

H transversal claimed

Level: Clifford · Cost: one parallel physical-H layer; self-duality swaps each paired X/Z face stabilizer and exchanges the declared logical X and Z representatives

References: bombin2006topological

S in the declared boundary-logical convention transversal claimed

Level: Clifford · Cost: one parallel layer with physical S on even row-major qubit labels and S-dagger on odd labels; every face contains equal bipartition counts, while the odd left-boundary representative maps X_L to i X_L Z_L; reversing the physical pattern implements the logical dagger

References: bombin2006topological

CNOT between aligned patches transversal claimed

Level: Clifford · Cost: one pairwise physical-CNOT layer between corresponding data qubits; alignment, routing, syndrome extraction, and recovery are additional

References: bombin2006topological

joint logical Pauli measurements surgery claimed

Level: measurement-based Clifford interface · Cost: compatible color-boundary interfaces, an ancilla region, repeated joint checks, decoding, and feed-forward; the cited low-overhead layout mixes 6.6.6 and 4.8.8 regions

References: thomsen2024lowoverhead

non-Clifford gate by magic-state injection or distillation measurement-based claimed

Level: non-Clifford resource protocol · Cost: a prepared or distilled resource state, verification or acceptance, logical Pauli measurements, and classical feed-forward; the cited factory uses color-code lattice surgery rather than adding a native transversal non-Clifford gate to the isolated patch

References: lee2025magiccolor