For a BB memory, “the decoder” is not a function of \(H_X,H_Z\) alone. The static matrices, extraction circuit, detector graph, noise model, and inference algorithm form one experiment. This page keeps their interfaces separate.
With a perfect measurement, an X-type data error \(e_X\) produces the Z-check syndrome \(H_Ze_X\); a Z-type error \(e_Z\) produces \(H_Xe_Z\). A correction needs the same syndrome and may differ from the actual error by a stabilizer. Because BB Tanner graphs have many short cycles and the code is degenerate, the most likely physical error and the most likely logical coset need not be the same object.
This code-capacity problem is the clean input for the BP-OSD baseline of roffe2020decoding and its generalized-bicycle precursor panteleev2021degenerate.
Measuring one stabilizer with an ancilla requires preparation, a sequence of entangling gates, measurement, and reset. The order of those entangling gates matters. An ancilla Pauli introduced early can propagate through later gates to several data qubits. Such a correlated “hook” may align with a logical operator more efficiently than independent faults.
bravyi2024bicycle gives a seven-CNOT-layer extraction for its two-layer connectivity. The number seven is a property of that schedule and layout, not of every weight-six matrix. shaw2025morphing instead changes the code during the cycle, reducing the connectivity degree from six to five and using six CNOT rounds for a related family. The comparison changes both the hardware graph and the detector model.
The Atlas constructor exports only static check incidence. It intentionally does not manufacture a schedule from row support: support does not determine CNOT order, ancilla assignment, reset placement, or detector definitions.
With noisy repeated extraction, one does not decode raw syndrome bits in isolation. Detector events compare measurements across time so that a measurement fault produces a recognizable spacetime pattern. The boundary rounds, initialization, final data measurement, and which products of measurement results are deterministic all enter the detector definition.
This leads to three distances that must not be conflated:
| object | minimal event |
|---|---|
| static code distance | nontrivial logical Pauli on data |
| circuit distance | undetected fault set causing a logical failure |
| decoder-found failure weight | smallest failure found under a specific search or decoder |
A static distance of 12 is compatible with a smaller circuit distance under a bad schedule. Conversely, observing no low-weight decoder failures does not prove circuit distance.
Belief propagation passes soft likelihood messages between error variables and parity constraints. On a tree, those local messages are exact. BB Tanner graphs are not trees: short cycles feed correlated information back into itself, and quantum degeneracy means many errors represent the same logical coset. BP may oscillate, converge to an inconsistent hard decision, or settle on the wrong coset.
Ordered-statistics decoding uses the BP reliabilities to choose an information set, then solves the syndrome constraints while testing a controlled set of reliability-ordered deviations. Its contribution is not another syndrome formula; it is a global repair stage when local messages fail.
The operational contract must specify at least:
Without these, “decoded with BP-OSD” is incomplete.
The card records the literature statement from bravyi2024bicycle: finite-size scaling across the paper's BB memory family gives an estimated circuit-level threshold of 0.8% under its standard circuit-noise model, depth-seven syndrome extraction, detector construction, and BP-OSD decoder.
This is not:
It is also not a one-block pseudo-threshold against a physical baseline. Changing the scaling family, circuit, noise normalization, or decoder defines a different threshold experiment and can move or remove the crossing. The qualification is part of the number's meaning.
The literature sweep exposes several ways to use more BB structure:
koutsioumpas2025autdec runs an ensemble over code automorphisms and combines candidates. The gain comes from moving a hard syndrome into different symmetry frames.sahay2026matching reduces selected BB problems to toric-like cylinders, applies matching, and combines it with BP and over-matching. The reduction's symmetry assumptions are part of the decoder.blue2026machine trains a transformer on circuit-level histories. Its training distribution, calibration, and inference latency are part of the claim.pakhunov2026convergence predicts when BP is likely to converge, acting as a triage layer rather than a complete correction rule.pakhunov2026peeling analyzes a greedy peeling/streaming route and reports large latency gains for its specified benchmarks. Those gains do not transfer without the same channel, implementation, and accuracy target.The card includes only decoder identifiers already registered in the shared vocabulary. The deeper page records newer compound pipelines without pretending each is the same algorithm.
The static entry is complete as a code description, but it does not yet ship a BB Schedule:
Adding those requires an engine-supported circuit/schedule representation and a source-faithful transcription. Until then, the generic lattice explorer shows exactly what it owns—data coordinates and stabilizer supports—and no more.
Static syndrome algebra is recomputed from the constructor. BP-OSD behavior and the seven-layer circuit/threshold estimate are literature claims from roffe2020decoding, panteleev2021degenerate, and bravyi2024bicycle. Morphing, automorphism ensembles, matching, learned decoding, convergence prediction, and peeling are scoped to their respective registered sources.