Question. Which property beyond CSS commutation makes a bitwise T-type phase preserve this code, and why does the answer contain a dagger?
CSS commutation tests overlaps between an X-check row and a Z-check row. Diagonal phase gates instead test how X-stabilizer codewords overlap with one another. The relevant five-row matrix is
G = [1; H_X],
where the first row is the odd all-one logical-X representative and the last four rows are even X stabilizers.
Index the rows by g_0=1 and the four coordinate functions g_1,...,g_4.
|g_0 and g_i|=8: a coordinate function is one at eight labels.|g_i and g_j|=4 for distinct coordinates: fixing two bits to one leaves four choices for the other bits.|g_0 and g_i and g_j|=4: intersecting with the all-one row changes nothing.|g_i and g_j and g_k|=2 for distinct coordinates: fixing three bits leaves two choices for the fourth.All pair and triple overlaps are even. This is the triorthogonality condition of Bravyi and Haah. The Atlas predicate transversal_t_candidate(H_X, LOGICAL_X) recomputes all ten pair tests and all ten triple tests on the shipped matrix.
Triorthogonality is stronger than the pairwise CSS relation. The Steane matrix, for example, has an odd three-row intersection and fails this test despite defining a valid self-dual CSS code.
The X-stabilizer space L contains the zero word and the fifteen nonzero linear functions. Their exact weight enumerator is
W_L(y) = 1 + 15 y^8.
Define the encoded computational basis using the X-stabilizer superpositions
|0_L> = (1/4) sum_(x in L) |x>
and
|1_L> = X(1)|0_L> = (1/4) sum_(x in L) |1+x>.
Every component of |0_L> has weight zero modulo eight. Hence bitwise physical T, whose phase on |x> is exp(i pi |x|/4), assigns the common phase one to the entire logical-zero state.
Complementing the zero word gives weight fifteen; complementing any weight-eight word gives weight seven. Therefore
W_(1+L)(y) = y^15 + 15 y^7,
and every logical-one component has weight seven modulo eight. Physical T^otimes 15 supplies the common phase
exp(i pi 7/4) = exp(-i pi/4).
In the ordered basis (|0_L>,|1_L>), the induced operation is therefore
diag(1, exp(-i pi/4)) = T^dagger.
The inverse physical layer, (T^dagger)^otimes 15, implements logical T. This derivation fixes the card's convention. Replacing a bare statement “transversal T” with this phase calculation prevents two sources using conjugate magic-state conventions from appearing to disagree.
The sign-free binary symplectic matrix establishes supports, commutation, and logical Pauli quotients. It does not retain the eighth-root phases introduced by T. That is why the reproduction transcript explicitly enumerates computational-basis weights in addition to running a triorthogonality predicate.
An XP-stabilizer representation can keep this phase data in one algebraic object; Webster, Brown, and Bartlett provide that broader formalism. The Atlas still treats XP as a curated dictionary here because the ordinary constructor exports a Pauli CheckMatrix, not a phase-aware XP object.
CNOT. Pairwise physical CNOT between identically ordered CSS blocks preserves the two stabilizer groups and induces logical CNOT. The support conjugation is exact at the check-matrix level; routing and recovery are separate circuit costs.
Hadamard. Bitwise H exchanges X and Z. Here their ranks are four and ten, so H cannot map the unchanged stabilizer group to itself. The card therefore does not call H transversal. Its cited H route switches to the Steane code, uses Steane's transversal H, then switches back.
Universality. T-type phase plus CNOT is not a complete native fault-tolerant instruction set for one unchanged block. The cited universal constructions add error correction, code switching, state preparation, measurement, or another encoded block. The protocol ledger accounts for those resources instead of charging them to transversality.
The pair/triple overlaps and logical diagonal phase are fully recomputed for this fixed constructor. The operations facet remains literature-originated because the repository does not yet certify complete logical gate sets, fault propagation, or circuit gadgets. That larger capability is tracked by wanted:transversal-gate-cert; the current exact proxy is the narrow triorthogonality and coset-phase calculation used here.