Logical operations

H transversal claimed

Level: Clifford · Cost: one parallel physical-H layer; it exchanges the identical X and Z face incidence matrices and swaps the declared logical X/Z boundary strings

References: bombin2006topological

S or S-dagger (logical convention dependent) transversal claimed

Level: Clifford · Cost: one parallel physical phase layer with S/S-dagger selected by the bipartition; the induced logical convention must be tracked for the chosen boundary basis

References: landahl2011color

CNOT between aligned patches transversal claimed

Level: Clifford · Cost: one pairwise physical-CNOT layer between corresponding qubits; geometric alignment, transport, syndrome extraction, and correlated recovery are additional resources

References: landahl2011color

joint logical Pauli measurements surgery claimed

Level: measurement-based Clifford interface · Cost: compatible colored boundaries, an ancillary region, repeated joint checks, decoding, and feed-forward; the logical action depends on the complete merge/split protocol

References: landahl2014latticesurgery

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

Level: non-Clifford resource protocol · Cost: encoded resource blocks, transversal inter-block Clifford operations, stabilizer/factory postselection or correction, measurement, and classical feed-forward

References: salesrodriguez2025magic