Question. Which algebraic feature makes H, phase, and inter-block CNOT preserve the code, and what logical gate does each physical layer induce?
Use X(h) and Z(h) for Paulis with binary support h. Let D=row(H)=C^perp be the simplex check space and choose ell=1110000 for the nontrivial coset in C/D.
A one-block transversal gate is a tensor product of one-qubit gates. An inter-block transversal gate couples qubit j of one block only to qubit j of another. A physical fault therefore does not spread from one location to multiple locations inside the same code block during that layer.
This is a fault-containment property, not a complete fault-tolerance proof. State preparation, syndrome extraction, routing, leakage, and the correction after the layer remain part of the gadget.
Single-qubit Hadamard conjugates X to Z and Z to X. Hence
H^⊗7 X(h) H^⊗7 = Z(h),
H^⊗7 Z(h) H^⊗7 = X(h).Because Steane uses the same space D for X and Z checks, each stabilizer is mapped to its paired stabilizer. The group is preserved.
For the chosen logical representatives,
X_L = X(ell) -> Z(ell) = Z_L,
Z_L = Z(ell) -> X(ell) = X_L.The induced logical gate is H. A generic CSS code need not have equal check spaces; bitwise H then maps it to the CSS code with the two spaces exchanged rather than acting within the original code.
Single-qubit S obeys
S X S^dagger = Y = i X Z and S Z S^dagger = Z.
On a support h,
S^⊗7 X(h) (S†)^⊗7 = i^|h| X(h)Z(h).
Every nonzero h in the simplex space has weight four. Therefore the phase is i^4=1, and both X(h) and Z(h) are stabilizers. Z stabilizers are fixed. The stabilizer group is preserved.
The chosen logical support has weight three:
X_L -> i^3 X_L Z_L = -i X_L Z_L = -Y_L,
where Y_L=i X_L Z_L. Conjugation X_L -> -Y_L is logical S^dagger, not logical S. Thus:
S^dagger;S^dagger on every qubit implements logical S.Some summaries call both statements “transversal S” because inverses are equally available. This entry retains the conjugation convention so the sign is auditable.
Apply CNOT from every qubit of a control Steane block to the corresponding qubit of a target block. Its Pauli conjugation is
X(h) ⊗ I -> X(h) ⊗ X(h),
I ⊗ X(h) -> I ⊗ X(h),
Z(h) ⊗ I -> Z(h) ⊗ I,
I ⊗ Z(h) -> Z(h) ⊗ Z(h).For h in D, every image is a product of stabilizers of the two blocks. For h=ell, the same formulas are precisely logical CNOT's action on X_L and Z_L. One faulty physical CNOT can create one error in each block, but it cannot create two data errors inside either block during this layer.
The abstract layer still assumes corresponding pairs can interact. Ion transport, all-to-all couplings, or movable neutral atoms can supply that matching directly; a fixed sparse array may pay routing depth and acquire additional fault locations.
The reproduction script verifies the common X/Z matrix, double-even simplex weights, logical representatives, and their commutation. Those facts make the conjugations above executable algebra.
The claim that a complete logical gadget remains fault tolerant under a specified circuit noise model is literature provenance, beginning with Shor's fault-tolerance framework. The absence of a universal transversal set for one additive code is placed in context by Zeng, Cross, and Chuang. Neither citation substitutes for the stabilizer calculations on this page.