Write a Pauli error as binary vectors (eX,eZ) on the edge qubits. For the CSS checks,
star syndrome sX = HX eZ,
plaquette syndrome sZ = HZ eX,with arithmetic modulo two. A single Z edge has the two endpoint vertices in sX. Along a longer connected path, every internal vertex touches two error edges and cancels; only the two endpoints remain. The dual statement holds for an X path and plaquette defects.
Suppose an error chain e and a candidate correction c have the same boundary. Their sum e+c is closed. Three cases matter:
e+c bounds a collection of faces. It is a stabilizer, so the correction succeeds.e+c has horizontal or vertical winding. It is a logical operator, so the correction returns to the code space but changes encoded information.e+c is not a purely spatial chain at all; its endpoints and time boundary must be evaluated in the spacetime detector complex.Thus decoding is a choice of logical class consistent with the data, not an instruction to find any path between visible endpoints.
Minimum-weight perfect matching assigns additive costs to graph edges, pairs defects, and chooses one minimum-cost representative. It is efficient and geometry-aware, but it does not sum all degenerate representatives in a class. Maximum-likelihood decoding compares the total probability of logical classes; in the statistical picture this is a free-energy rather than ground-energy decision.
Renormalization-group decoders approximate class probabilities while merging scales. Tensor-network methods contract a class partition function exactly for restricted channels or approximately at controlled bond dimension. BP-OSD uses generic sparse-matrix soft information and post-processing; it does not automatically exploit periodic homology. A cellular automaton restricts communication to a finite light cone and therefore solves an additional locality problem.
| Regime | Decoder receives | Added ambiguity | Claim boundary |
|---|---|---|---|
| Code capacity | one perfect spatial syndrome and a Pauli prior | which spatial homology class | No measurement or circuit faults |
| Phenomenological | syndrome changes over repeated rounds | paths may run in time | Round count, time boundary, and data/check normalization matter |
| Circuit level | detector events from a named extraction circuit | correlated hooks, leakage, and schedule faults | A threshold belongs to that circuit and decoder |
| Detectable loss | syndrome plus known missing-edge locations | supercheck geometry and percolation | Known erasure is not unheralded leakage |
| Correlated Pauli | syndrome plus a joint prior | paths no longer have independent edge weights | Matching needs reweighting or a richer objective |
| Coherent rotation | amplitudes or a channel model, not only Pauli probabilities | interference between fault histories | Pauli twirling can change the logical channel |
reproduce.py applies consecutive Z errors along one horizontal row. Every proper open segment has exactly two star defects. Extending the segment by one edge moves one endpoint. Completing all L edges closes a noncontractible loop: the syndrome vanishes, yet the vector is outside the stabilizer row space and equals a constructor-provided logical support.
This small experiment is the whole failure mechanism in miniature. The decoder sees the same zero syndrome for identity, any stabilizer, and any closed logical loop. Only the prior and error history distinguish them.
The threshold ledger records source-reported values only with channel, syndrome quality, decoder, and observable. It does not transfer an open-Surface circuit number to the torus or compare a detectable-loss percolation point with an unheralded Pauli rate.
Primary sources: dennis2002topological, wang2003confinement, wang2010thresholds, stace2009loss, duclos2010fast, bravyi2014efficient, breuckmann2017local, chubb2021statistical, and roffe2020decoding.