This page first derives ideal Pauli conjugation for the exact constructor presentation. Fault-tolerant circuits, routing, repeated extraction, and logical error rates are separate protocol layers.
Physical H conjugates X_q <-> Z_q on every qubit. Since every face support appears once in each CSS sector,
H^(tensor n): X_f <-> Z_f.
The stabilizer group is preserved exactly. The declared boundary operators also swap, X_L <-> Z_L, so the logical action in this basis is H. The transversal statement follows from H_X=H_Z; it does not require rotating the triangular patch afterward.
Join consecutive vertices around each retained face. The resulting physical vertex graph is bipartite. Under the constructor's row-major qubit order, the two parts are precisely even labels A and odd labels B. Every weight-four face contains two vertices from each part, and every weight-six face contains three from each.
Apply
U_S = product_(q in A) S_q · product_(q in B) S_q^dagger.
Single-qubit conjugation gives
S X S^dagger = i XZ,
S^dagger X S = -i XZ,
while Z is fixed. If a face contains a vertices in each part, its X check acquires phase i^a(-i)^a=1 and a Z on every support qubit. Therefore
U_S X_f U_S^dagger = X_f Z_f,
U_S Z_f U_S^dagger = Z_f.
The left boundary alternates between the two parts and begins and ends in A. It contains (d+1)/2 A sites and (d-1)/2 B sites. Its conjugation phase is
i^d (-1)^((d-1)/2)=i,
so X_L -> i X_L Z_L and Z_L -> Z_L: logical S in the declared basis. Swapping the physical S and S-dagger layers implements the inverse. The reproduction transcript checks every face balance and logical phase for d=3,5,7 rather than inferring them from a diagram.
For a physical CNOT with control c and target t,
X_c -> X_c X_t, Z_c -> Z_c,
X_t -> X_t, Z_t -> Z_c Z_t.
Apply this pairwise to two copies with identical qubit order. A control X-face check gains the corresponding target X-face check; a target Z-face check gains the corresponding control Z-face check. Control Z and target X checks stay in their original blocks. Thus the two-block stabilizer group is preserved. Replacing a face support by the left-boundary support gives the logical CNOT conjugation on X_L and Z_L.
"Transversal" here means one pairwise layer after corresponding qubits can interact. A planar device may still need movement, routing, or a stacked layout, and a fault-tolerant operation still needs syndrome extraction and recovery around the layer.
H, S, and CNOT generate Clifford computation. They are valuable because the physical action is unusually direct, but they are not universal. The Bravyi–König theorem places every bounded-spread constant-depth local logical gate in a 2D growing- distance topological stabilizer family at the second level of the Clifford hierarchy. T lies at the third level.
This is not a statement that all non-Clifford protocols are impossible. It identifies what extra ingredient they must use:
The hybrid 6.6.6–4.8.8 surgery architecture of Thomsen et al. and the color-code magic- state factory of Lee et al. retain these resource boundaries. Their overhead is not another row in the static check matrix. Flag-assisted preparation in Chamberland and Noh likewise includes redundant encoding, acceptance, and verification rather than turning a bare transversal Clifford layer into a non-Clifford gate.
The executable transcript proves the ideal Pauli maps for the shipped basis. The card still marks operations as claimed because the shared verifier does not yet certify logical gates with phase, stabilizer equivalence, geometry, multi-block alignment, and fault model all explicit. That general capability is routed to wanted:transversal-gate-cert; local algebra evidence is not silently promoted into gadget-level verification.