Every row below describes the same code only within a stated domain. A useful dictionary names both what it preserves and what it changes.
| picture | preserves | adds insight | must not be inferred |
|---|---|---|---|
| sparse CSS incidence | exact checks, ranks, logical quotient | machine-verifiable finite algebra | boundary physics not encoded by a matrix name alone |
| colored homology | stabilizer deformation classes and boundary endpoints | why three string colors describe one logical pair | a numerical decoder or circuit threshold |
| anyon condensation | excitation fusion and boundary absorption sectors | precise color/Pauli boundary and surgery language | that equally colored drawn edges are identical interfaces |
| folded surface codes | logical phase under local-Clifford equivalence | branching strings become attached surface strings | exact check/qubit order for this constructor |
| random three-body Ising | perfect-syndrome likelihood classes | three-way error correlations and phase transition | an efficient decoder or a circuit-level result |
| tensor network | local encoder tensors and operator pushing | how patches may be composed | inherited global parameters after contraction |
The homology source is Bombín and Martín-Delgado, the unfolding dictionary is Kubica, Yoshida, and Pastawski, the statistical model is Katzgraber, Bombín, and Martín-Delgado, the boundary classification is Kesselring et al., and the tensor construction is Farrelly, Tuckett, and Stace.
6.6.6 versus 4.8.8. Both realize the abstract two-dimensional color-code order on trivalent, face-three-colorable cellulations. Their face weights, vertex density, clipped boundaries, extraction circuits, and finite qubit formulas differ. The comparative resource analysis of Bombín and Martín-Delgado supports the relation, not numerical transfer between entries. In particular, the circuit study of Landahl, Anderson, and Rice analyzes a 4.8.8 architecture; its numerical results are context, not 6.6.6 data.
Triangular color boundary versus Pauli boundary. The constructor fixes one red, green, and blue side. A patch whose sides condense Pauli-labelled sectors can have a similar outline and different logical/surgery behavior. The boundary condition belongs in the formal identity.
Seven-qubit Steane block. The distance-three member is locally the same stabilizer code up to row and qubit relabeling. Steane's algebraic presentation and this family's growing geometry answer different questions; fixed-block circuit evidence is not automatically scaling evidence.
Hybrid lattice surgery. Joining 6.6.6 and 4.8.8 regions deliberately changes the local tiling to trade face weight against layout cost. The hybrid device is neither an alias nor an isolated member of either uniform family.
Tensor contraction. Connecting encoder tensors removes or pairs legs and creates a new global stabilizer code. Every resulting n, k, boundary, and distance must be recomputed.
| platform evidence | established | did not establish |
|---|---|---|
| seven-qubit trapped-ion block | encoded operations on the smallest triangular color/Steane member | increasing-distance family suppression |
| trapped-ion fault-tolerant logical processor | Clifford primitives combined with magic-state preparation and injection | a universal transversal set on one 2D block |
| reconfigurable neutral-atom architecture | many small logical blocks and transversal inter-block operations | growth of one 6.6.6 memory patch |
| superconducting triangular processor | two finite sizes, repeated extraction, hardware-specific decoding, Clifford gates, injection, and logical teleportation | an extraction-independent threshold or asymptotic family constant |
The corresponding primary records are Nigg et al., Postler et al., Bluvstein et al., and Lacroix et al..
As of the 2026-08-02 realization sweep, no photonic 6.6.6 demonstration met the Atlas distinction between a proposal and an executed logical code. The same sweep found that only the superconducting record in this set directly compared two color-code sizes; the earlier results remain important fixed-block or multi-block evidence.
An equivalence is most useful when it tells the reader which computation to move to another language. Unfolding motivates projected surface-code decoding; anyon condensation organizes boundary joins; the Ising model organizes likelihood correlations; tensor pushing organizes composition. None authorizes copying a distance, threshold, or hardware number without rebuilding the finite identity tuple. That restriction is the difference between a representation dictionary and an accidental conflation of codes.