Logical operations

logical X / Z Pauli loops transversal verified

Level: Pauli · Cost: one parallel physical-Pauli layer on a noncontractible length-L support, or a classical Pauli-frame update

CNOT between corresponding logical sectors of two aligned Toric blocks transversal verified

Level: Clifford · Cost: one pairwise physical-CNOT layer; alignment, transport, and correlated decoding are external architecture costs

single-block Clifford generating set fold-transversal claimed

Level: Clifford · Cost: fold pairings, Dehn-twist deformation, and single-shot logical Pauli measurements with protocol-specific ancillas and decoding

References: guernut2024toric

logical Pauli measurement measurement-based claimed

Level: stabilizer measurement · Cost: single-shot measurement gadget plus an ancilla layout and decoder from the proposed Toric Clifford protocol

References: guernut2024toric