Question. Why does the same weight structure that permits a transversal phase gate reject T, and which extra resource repairs the missing gate?
The answer has a code-internal part—the obstruction—and a protocol part—the choice of an external non-Clifford route. They must not be merged into the false sentence “Steane has universal transversal gates.”
Write the logical zero as
|0_L> = (1/sqrt(8)) sum_{u in C^perp} |u>.
The simplex space contains the zero word and seven words of weight four. Bitwise T acts on a computational word u with phase exp(i pi |u|/4). The zero word receives phase +1, while every weight-four word receives phase -1.
The resulting state is supported only on the C^perp coset. Any code state with support only on that coset must be proportional to |0_L>, because |1_L> occupies the disjoint coset C^perp+ell. The coefficients are not uniform after bitwise T, so the result is not in the code space. This is stronger than saying the logical action is the wrong T: the physical layer fails to preserve the code.
Bitwise S asks only for the stabilizer weights to vanish modulo four, which they do. Bitwise T distinguishes weight zero from weight four, so the finer phase condition fails.
The three X-check supports are
{3,4,5,6}, {1,2,5,6}, {0,2,4,6}.Every pair overlaps evenly, which is enough for CSS commutation. All three overlap at qubit 6, so their triple overlap is odd. The Atlas transversal_t_candidate check appends an odd-weight logical-X row and tests the Bravyi–Haah triorthogonality conditions; it returns false, with this three-check overlap as the obstruction in the shipped basis.
Pairwise orthogonality and triorthogonality answer different questions. The Steane matrix is a compact example where the former holds and the latter fails.
| route | imported resource | fault-containment idea | cost that must remain visible |
|---|---|---|---|
| Magic-state injection | a prepared non-stabilizer logical state | keep the online data circuit Clifford and consume the state through measurement/feed-forward | state preparation or distillation, verification/postselection, and classical control |
| Steane/Reed–Muller switching | a second code with transversal non-Clifford phase | convert fault tolerantly, apply the partner's transversal gate, correct, and convert back | a fifteen-qubit block, conversion syndromes, extra correction rounds, and switching failure modes |
| Pieceable CCZ | intermediate error correction inside a non-transversal multi-block circuit | interrupt propagation before correlated faults become uncorrectable | several circuit pieces, intermediate syndromes, and a decoder aware of correlated errors |
| Divisible-code concatenation | an outer divisible CSS block | move the required phase divisibility into the enlarged code | substantially larger blocks and concatenated preparation/recovery |
Injection. Trapped-ion work combined fault-tolerant Steane Cliffords with logical magic-state preparation and injection (Postler et al.). Universality is a property of that state-preparation and consumption protocol.
Switching. Paetznick and Reichardt combine transversal gates with error correction across code presentations (paper). Anderson, Duclos-Cianci, and Poulin give explicit fault-tolerant conversion between Steane and the fifteen-qubit quantum Reed–Muller code and interpret the pair as gauges of a larger subsystem code (paper). A later trapped-ion experiment converted and certified a logical magic state (Daguerre et al.).
Pieceable CCZ. Yoder, Takagi, and Chuang interrupt a non-transversal CCZ with intermediate correction so dangerous propagation does not accumulate through the entire circuit (paper). This is not a transversal Steane gate; the pieces and intervening recovery are the mechanism.
Divisible concatenation. Divisible-code constructions embed the seven Steane qubits into a larger CSS block whose weight divisibility supports the desired phase action (Hu, Liang, and Calderbank). The larger block is the resource, so its overhead cannot be charged to the bare [[7,1,3]] code.
card.yaml records switching because the current operation vocabulary has a switching mechanism. It does not force pieceable CCZ or divisible concatenation into an inaccurate existing enum. Adding pieceable and concatenation operation mechanisms requires a schema task outside T12's atlas/steane/ scope. This page preserves the knowledge while keeping the structured gap honest.
The card's switchable_with target is intentionally pending until the fifteen-qubit Reed–Muller entry exists. A dangling relation is more truthful than pretending the switching partner has already been authored.
The fetch-verified operations sweep found no source giving a universal transversal gate set on one Steane block. Every qualifying universal proposal or experiment imports at least one of the resources above. The general non-universality context is transversality versus universality; the page-level algebra above identifies the concrete T failure for this code.