A threshold is a transition of a family of experiments, not an invariant of a finite check matrix. Its identity tuple is
(family and boundary, physical channel, syndrome/circuit model,
decoder and objective, rate normalization, asymptotic observable).Changing any coordinate can change the value. The Toric card therefore names decoder regimes without putting one number in the entry summary.
Every value below has origin {kind: claimed, ref: BIBKEY}. None is recomputed by the Atlas constructor.
| Source-reported value | Identity tuple | What may be concluded |
|---|---|---|
15.5% | Periodic Toric family; ideal syndrome extraction; each data qubit receives X, Y, or Z with probability p0/3; graph matching; the paper converts the X-component rate to p=(2/3)p0; asymptotic logical transition | A benchmark in the paper's total single-qubit depolarizing-rate convention, not the independent-X/Z maximum-likelihood value from another convention. Origin: wang2010thresholds. |
7.8 × 10^-3 | Periodic Toric family; the named four-interaction-order extraction circuit; preparation, readout, memory, and two-qubit processes each assigned rate p, with the fifteen nonidentity two-qubit Paulis equiprobable after an ideal CNOT; repeated graph matching; asymptotic memory transition | A circuit-level result in that paper's equal-process-rate units, not a code-capacity or generic modern circuit threshold. Origin: wang2010thresholds. |
18.9(3)% | Toric code; iid single-qubit depolarizing channel; correlations between X, Y, and Z components retained through an eight-vertex-model mapping; optimal-class transition | A depolarizing-channel threshold under the paper's correlated Pauli convention, not three independent binary matching problems. Origin: bombin2012strong. |
50% detectable loss | Square-lattice topological code; erasure locations known; loss geometry controlled by bond percolation, with computational-error tradeoff treated separately | The maximum tolerable heralded-loss boundary of that model, not an unheralded leakage rate. Origin: stace2009loss. |
0.133% | Two-dimensional Toric code; equal independent data and syndrome error probabilities; Harrington cellular automaton with finite communication speed | A local-decoder benchmark under a restrictive symmetric phenomenological model, not the capability of global matching. Origin: breuckmann2017local. |
These rows are deliberately heterogeneous. Their purpose is to prevent a reader from ranking decoders or hardware using incomparable scalars.
Coherent information can exhibit an information-theoretic transition without choosing a decoder. lee2025coherent derives an exact Toric coherent-information expression and links it to random-bond Ising criticality. That result addresses recoverable information, not the runtime or decision rule of MWPM, tensor contraction, or BP-OSD.
Coherent rotations also carry phase information absent from a stochastic Pauli channel. bravyi2018coherent gives an efficient Majorana simulation for declared coherent-error models; iverson2020coherence proves a conditional logical-incoherence trend and explicitly does not extend it to fixed physical noise strength as L grows. Neither result licenses replacing coherent noise by a universal Pauli rate.
A finite logical error per round, a two-distance comparison, a prepared-code state fidelity, and an asymptotic decoder threshold answer different questions. The 2026 neutral-atom Toric experiment reports repeated extraction and a finite comparison; the Atlas records it under realizations and does not insert it into the table above. See operations and realizations.
10.1103/Physics.5.50 resolves to an APS Physics Viewpoint. It is secondary commentary and not the primary origin of a numerical row.10.1103/PhysRev.88.1332 and DOI 10.1143/JPSJ.55.3305 are classical Ising-model sources. They are precursors to the statistical machinery, not direct Toric-code threshold reports.1810.09621 concerns measurement-based fault tolerance beyond foliation; its placement near a threshold feature does not make it a Toric code-capacity benchmark.Primary sources: wang2010thresholds, bombin2012strong, stace2009loss, breuckmann2017local, lee2025coherent, bravyi2018coherent, and iverson2020coherence.