Entry facet
Logical operations
Available gates, mechanisms, costs, and provenance.
CNOT between aligned Bacon–Shor blocks transversal claimed
Level: Clifford · Cost: one pairwise physical-CNOT layer between matching grid sites; block alignment, routing, syndrome extraction, and recovery are additional
References: aliferis2007subsystem
H on a square block transversal claimed
Level: Clifford · Cost: one physical-H layer followed by the row/column permutation that restores the declared vertical-X and horizontal-Z gauge orientation
References: aliferis2007subsystem
multi-controlled Z family on dimension-matched rectangles transversal claimed
Level: non-Clifford controlled-phase hierarchy · Cost: block dimensions must satisfy the cited geometry for the chosen control order; changing aspect ratio and performing error correction between pieces are protocol resources, not free transversal gates on one square
References: yoder2017universalbacon
joint logical parity and state transfer surgery claimed
Level: measurement-based Clifford interface · Cost: a compatible boundary, an ancilla code patch, repeated joint gauge measurements, and classical feed-forward
References: nautrup2017interface
biased-noise universal primitives measurement-based claimed
Level: teleportation-assisted universal set · Cost: verified ancillas, teleportation measurements, and correction around strongly rectangular blocks; the aspect ratio is optimized jointly with the hardware noise bias
References: brooks2013asymmetric
CCZ by measurement-free deformation switching claimed
Level: non-Clifford · Cost: coherent correction circuits and extra qubits replace mid-circuit measurement while the code is deformed; measurement-free does not mean correction-free
References: veroni2025measurementfree
logical operations on the Floquet–Bacon–Shor descendant measurement-based claimed
Level: dynamical logical control (variant only) · Cost: a period-four measurement schedule, time-dependent logical tracking, and detector decoding; this operation is not attached to the static checkmatrix exported by this entry
References: sun2025floquetbacon