Logical operations

CNOT surgery claimed

Level: Clifford · Cost: O(d) rounds per merge/split plus an ancilla routing patch

References: horsman2012surface

CNOT transversal claimed

Level: Clifford · Cost: constant-depth physical CNOT layer; transport and correlated-decoding costs remain

References: bluvstein2024logical

CNOT braiding claimed

Level: Clifford · Cost: O(d) or more rounds and a distance-sized reserved hole trajectory

References: fowler2012surface

single-qubit Clifford braiding claimed

Level: Clifford · Cost: twist/corner code deformation over distance-preserving paths

References: brown2017poking

H / S fold-transversal claimed

Level: Clifford · Cost: constant-depth gates plus a fold or equivalent qubit permutation

References: moussa2016transversal

logical |0> / |+> initialization measurement-based claimed

Level: stabilizer preparation · Cost: prepare a physical product state, measure the complementary stabilizers for d rounds, and decode the spacetime record

References: fowler2012surface

logical Z / X readout measurement-based claimed

Level: stabilizer measurement · Cost: destructively measure all data qubits in the selected basis and decode the final data/syndrome history

References: fowler2012surface

Pauli-frame update measurement-based claimed

Level: Clifford bookkeeping · Cost: a classical frame update after decoded logical measurements; no immediate physical Pauli layer is required

References: fowler2012surface

T measurement-based claimed

Level: non-Clifford · Cost: inject and distill a magic state, then consume it by gate teleportation; factory spacetime dominates

References: bravyi2005magic