This page separates operations proved from the exported gauge algebra from protocols whose fault tolerance and resource cost remain literature claims.
Take two identically ordered r × c blocks, A as control and B as target, and apply physical CNOT pairwise at every grid site. Pauli conjugation gives
Apply these rules to each local generator:
Every image is in the two-block gauge group, so the center and protected subsystem are preserved. The bare logical row and column obey the same conjugation rules, giving logical CNOT in this card's Pauli-type convention. This establishes the abstract transversal layer. It does not include routing, faulty CNOT locations, extraction, or recovery.
Physical H swaps X and Z. On the declared grid it maps vertical X gauge edges to vertical Z edges and horizontal Z gauge edges to horizontal X edges, which does not match constructor.py's orientation. Transpose the grid coordinates after the H layer:
(i,j) -> (j,i).
Vertical edges become horizontal and horizontal edges become vertical. For a square block this restores the gauge-generator set and exchanges the logical X row with the logical Z column. A nonsquare rectangle is mapped to the transposed c × r block, so it is not an unchanged-block transversal H.
reproduce.py performs this support map on the 3 × 3 generator set and checks exact equality after the Pauli swap.
3 × 3 gauge fix to the Shor stabilizer spaceThe square Bacon–Shor center contains two X strips:
XXXXXX___ and ___XXXXXX.
Promote all six horizontal Z gauge pairs to stabilizers:
ZZ_______, _ZZ______, ___ZZ____, ____ZZ___, ______ZZ_, and _______ZZ.
These eight rows are exactly the set exported by atlas/shor_nine/, up to row order. The reproduction transcript loads both constructors and proves mutual row-space containment and equal rank; this is stronger than comparing the parameter label [[9,1,3]].
The stabilizer equality does not license silently copying logical names. This entry uses Pauli-type labels:
| Bacon–Shor name | representative | Shor entry's encoded-basis name |
|---|---|---|
| logical X | XXX______ | logical Z |
| logical Z | Z__Z__Z__ | logical X |
Both pairs generate the same protected Pauli algebra and anticommute once. The Shor entry names them by their action on Shor's chosen encoded cat-state basis, while this entry names them by physical Pauli type. The gauge fix preserves the protected algebra but the cross-entry dictionary includes a logical Hadamard relabeling.
Promoting a gauge operator also fixes one formerly disposable degree of freedom. A fault-tolerant gauge-fixing protocol must record the measured outcomes, any repeated checks, and the Pauli-frame update; row-space equality alone is not a circuit.
The full rectangle promotes all horizontal Z gauges in the Shor specialization. Other commuting promotion patterns produce two-dimensional compass-code descendants (li2019compass). The selected pattern changes stabilizer weights, exposed syndrome information, and decoding geometry. It cannot be represented by changing only rows and columns on this card, so the relation target is a separate pending family.
Bravyi–Bacon–Shor changes a different resource: a binary occupancy matrix chooses which grid sites exist. Its encoded dimension and distances are properties of row and column codes of that matrix. The full rectangular family here is a special case; generalized rate claims do not flow backward to this card.
| mechanism | what changes or is consumed | claim boundary |
|---|---|---|
| dimension-matched transversal controlled-Z family | rectangle dimensions and correction between protocol pieces | yoder2017universalbacon; not a universal transversal set on one fixed square |
| biased-noise teleportation | strong aspect ratio, verified ancillas, teleportation measurements, correction | brooks2013asymmetric; geometry alone is not the architecture |
| subsystem lattice surgery | compatible boundaries, ancilla code patch, repeated joint gauge measurements, feed-forward | nautrup2017interface; not pairwise block transversality |
| measurement-free deformation | coherent correction circuitry and extra qubits replace mid-circuit measurement | veroni2025measurementfree; measurement-free is not correction-free |
| Floquet logical control | period-four schedule, detector decoding, time-dependent logical tracking | sun2025floquetbacon; descendant only |
As of the 2026-08-02 operations sweep, no fetched source supports a universal transversal set on one unchanged square Bacon–Shor block. Universality in the cited protocols uses aspect-ratio changes, teleportation, deformation, concatenation, surgery, or error-correction steps between pieces.
The CNOT and square-H support maps are exact statements about ideal Pauli conjugation. Calling them fault tolerant additionally requires a fault set, propagation analysis, extraction schedule, decoder, and recovery placement. The remaining operations in the table are kept claimed on card.yaml because their resource and fault-tolerance statements come from named protocols rather than the static constructor.