The constructor exports two primal Z cycles and two dual X cycles. For each support it builds a binary symplectic row and verifies three conditions:
Applying one string is a logical Pauli. Because its single-qubit factors can be applied in parallel, the physical depth is one when the control hardware allows simultaneous Paulis. In practice a decoded logical Pauli is usually recorded in a classical Pauli frame, avoiding a physical layer.
The support has weight L; “depth one” does not mean “weight one” and does not remove control, calibration, or correlated-fault concerns.
Align two identical blocks and apply physical CNOT from each control edge to the corresponding target edge. Pauli conjugation acts as
Xc -> Xc Xt, Xt -> Xt,
Zc -> Zc, Zt -> Zc Zt.An X star on the control becomes the product of corresponding stars on both blocks; a target Z plaquette similarly becomes the product on both blocks. The other two stabilizer sectors remain in place, so the joint stabilizer group is preserved. Exactly the same mapping holds for each homological logical pair, implementing CNOT separately on both encoded qubits.
reproduce.py constructs the doubled symplectic stabilizer space, performs this conjugation, checks equality of row spaces, and checks every selected logical generator. This proves the code-space action. It does not prove that a given device can align blocks, suppress transport faults, or decode correlated two-block events.
guernut2024toric proposes and simulates a constant-depth Toric Clifford toolkit combining fold-transversal gates, Dehn twists, and single-shot logical Pauli measurements. These ingredients require geometric pairings, cellulation-deformation schedules, ancillas, and decoding assumptions absent from the static constructor. The card therefore marks the toolkit and logical measurement as {kind: claimed, ref: guernut2024toric}.
bravyi2013classification supplies a boundary rather than an implementation: locality-preserving constant-depth logical gates for two-dimensional topological stabilizer codes are restricted to the Clifford level. A non-Clifford protocol for a broader abelian phase may add twists or enriched structure, as in bauer2025nonclifford; it is not a native operation of the unmodified square Toric stabilizer. T144 consequently adds no bare T gate.
| Source | Demonstrated object | Quantitative identity | Explicit non-claim |
|---|---|---|---|
bluvstein2022processor | Neutral-atom preparation and characterization of a Toric-code state on periodic geometry | A twenty-four-qubit Toric state in the experiment's transported-atom processor | Not repeated syndrome extraction, arbitrary logical-state memory, or below-threshold scaling |
atomcomputing2026toric | Repeated mid-circuit Toric syndrome extraction with lost-atom replacement and reservoir reloading | Logical error characterized through as many as ninety cycles; two code distances compared only through eight rounds, with lower absolute logical error for the larger distance in that range | Not an asymptotic or model-independent threshold claim |
Both rows are demonstrated because the reported protocols ran on hardware. The descriptions remain different because demonstration status does not erase the distinction between state preparation and active error correction.
When encountering a Toric operation or experiment, ask in this order:
This sequence prevents a valid algebraic gate from becoming an unsupported hardware claim and prevents a finite experiment from becoming a universal threshold statement.
Primary sources: kitaev2003fault, bravyi2013classification, guernut2024toric, bauer2025nonclifford, bluvstein2022processor, and atomcomputing2026toric.