Logical operations

protected logical Pauli string transversal verified

Level: Pauli · Cost: one parallel physical-Pauli layer on all L data qubits; X in the bit-flip convention and Z in the phase-flip convention

CNOT between aligned blocks in the same basis convention transversal verified

Level: Clifford · Cost: one pairwise physical-CNOT layer between corresponding sites; routing, syndrome extraction, and correlated recovery remain additional

global Hadamard basis exchange switching verified

Level: Clifford basis change · Cost: one physical-H layer on every data qubit, exchanging neighboring ZZ and XX checks together with the protected and unprotected Pauli sectors

joint logical parity between split repetition blocks surgery claimed

Level: measurement-based Clifford operation · Cost: a surface-code-derived coupling region, merge/split syndrome rounds, ancilla measurements, correlated decoding, and feed-forward

References: besedin2025repetition