The code is one stabilizer subspace, but different representations make different invariants cheap to see. A responsible conversion states what is preserved and what extra assumptions are introduced.
| Picture | Native object | Makes immediate | Does not provide by itself |
|---|---|---|---|
| Symplectic CSS | binary HX, HZ | commutation, rank, finite syndrome, row-space logical tests | geometry, circuit, or likelihood |
| Chain complex | C2 -> C1 -> C0 | boundary cancellation, homology, primal/dual cycles | decoder probabilities or hardware schedule |
| Cyclic HGP | two square neighbor-parity maps | tensor-separated directions and two kernel sectors | identity with arbitrary HGP outputs |
| Anyon strings | charges, strings, intersections | endpoint motion, fusion, logical commutation | a finite check ordering or decoder runtime |
| Statistical mechanics | disorder sectors and partition functions | class likelihoods and phase transitions | one model-free threshold |
| PEPS | local tensors with virtual symmetry | topological degeneracy and local indistinguishability | active QEC or repeated syndrome circuits |
| Majorana simulation | quadratic fermionic evolution in restricted channels | efficient coherent-error calculation | a physical tetron realization or arbitrary coherent-noise decoder |
The constructor takes the face boundary d2 and edge boundary d1, then emits HX=d1 and HZ=d2^T. This conversion preserves the exact qubit and check order. The identity d1 d2=0 becomes HX HZ^T=0; chain cycles become commuting Pauli rows, and chain boundaries become stabilizer products.
The symplectic picture forgets why the matrices commute but is ideal for machine checks. The chain picture restores the geometric reason and gives the quotient ker/im that classifies logicals.
The cyclic-HGP converter preserves the ordered matrices only for the stated orientation HGP(C_L,C_L^T). Open repetition factors change dimensions and kernel sectors, producing the planar Surface witness. General HGP factors can produce codes with no two-dimensional local embedding. The exact dictionary therefore supports a typed construction relation, not a claim that Toric and HGP are synonyms. See cyclic HGP.
The anyon dictionary turns a violated star or plaquette into an endpoint charge. Local Pauli errors create pairs; extending a string moves an endpoint; closing and winding the string performs a logical. It preserves fusion and intersection information but suppresses the row ordering used by software.
The statistical dictionary fixes a syndrome and groups compatible error chains by homology. A disorder-sector partition function sums the probability of a class. The critical point depends on the channel and syndrome model, so the representation explains thresholds only after the model tuple is fixed. dennis2002topological supplies the direct QEC mapping, while earlier Ising papers remain classical precursors.
In a PEPS representation, a virtual Z2 symmetry makes distinct torus sectors locally indistinguishable and produces topological degeneracy [schuch2010peps]. This is a ground-space representation. It does not imply that preparing the PEPS, measuring syndromes fault tolerantly, and decoding a memory are the same task.
For selected coherent channels, the correction calculation maps to free Majorana fermions and admits efficient simulation [bravyi2018coherent]. iverson2020coherence gives a conditional logical-channel result and states the fixed-noise-strength limitation. The mapping is therefore curated with the majorana-pairs dialect and routed to wanted:fermionic-compiler; the Atlas has no general executable converter for this view.
Sharing a representation or bulk phase is weaker than code identity. The open Surface entry shares star/plaquette bulk but changes boundary homology. The Honeycomb protocol has Toric-equivalent instantaneous frames but remains a periodic measurement process with an anyon automorphism. A broad Z2 topological phase can share anyon data while using a different microscopic Hamiltonian or qubit layout. The Toric card keeps these as typed relations and never expands its identity to absorb them.
Primary sources: kitaev2003fault, dennis2002topological, tillich2009quantum, schuch2010peps, bravyi2018coherent, iverson2020coherence, hastings2021dynamically, and aasen2022automorphism.