Decoders

Minimum-weight perfect matching mwpm standard

Represents detection events as vertices of a weighted graph and pairs them by minimum total weight; the pairing determines a likely correction chain.

Type: matching · Noise setting: independent X/Z noise with perfect syndrome (code capacity)

For this code: Matching reaches about 10.3%, while maximum-likelihood decoding reaches about 10.9% under the corresponding independent-noise convention. The gap is path entropy: matching minimizes chain energy rather than class free energy.

References: higgott2022pymatching, dennis2002topological, wang2003confinement

Minimum-weight perfect matching mwpm standard

Represents detection events as vertices of a weighted graph and pairs them by minimum total weight; the pairing determines a likely correction chain.

Type: matching · Noise setting: phenomenological data and syndrome noise over repeated rounds

For this code: The spacetime matching transition is about 2.9% for the symmetric phenomenological model studied by Wang, Harrington, and Preskill. This number is not a circuit-level gate threshold.

References: higgott2022pymatching, wang2003confinement

Minimum-weight perfect matching mwpm standard

Represents detection events as vertices of a weighted graph and pairs them by minimum total weight; the pairing determines a likely correction chain.

Type: matching · Noise setting: circuit-level depolarizing syndrome-extraction circuit

For this code: Fowler's scheduled eight-step syndrome circuit has a threshold near 0.57% per circuit step, or roughly 4% per full cycle in that paper's stochastic model. The value is schedule-, gate-set-, and decoder-dependent, so it must not be compared directly with the code-capacity figures above.

References: higgott2022pymatching, fowler2012surface

Union-find union-find standard

Grows and merges syndrome clusters until they can be neutralized, trading some matching accuracy for near-linear-time decoding.

Type: union-find · Noise setting: perfect syndrome and faulty repeated syndrome benchmarks

For this code: The toric-bulk benchmarks give about 9.9% with perfect syndrome and 2.6% with faulty syndrome measurements. Union-find trades a small threshold loss relative to matching for almost-linear worst-case processing.

References: delfosse2021almost

Renormalization-group decoder renormalization-group standard

Coarse-grains syndrome and error distributions hierarchically, using local inference to retain degeneracy information that a single lightest-path calculation discards.

Type: renormalization-group · Noise setting: code-capacity Pauli noise on surface/toric-code lattices

For this code: Hierarchical coarse-graining estimates the probability of logical classes rather than committing immediately to one globally lightest chain. It exposes a distinct speed/accuracy tradeoff and generalizes naturally to other topological-code lattices.

References: duclos2010fast

Tensor-network maximum-likelihood decoder tensor-network-mld standard

Compares logical-class probabilities by exact free-fermion methods for restricted noise or by approximate tensor-network contraction for more general channels.

Type: tensor-network · Noise setting: independent X/Z noise exactly; general Pauli noise approximately

For this code: Exact free-fermion contraction is available for the restricted independent channel, while matrix-product-state contraction approximates maximum likelihood for more general noise. The bond dimension controls the accuracy/runtime tradeoff.

References: bravyi2014efficient

BP + ordered-statistics bp-osd standard

Uses belief propagation for soft error estimates, then ordered-statistics post-processing to repair failures caused by short cycles and degeneracy.

Type: bp-osd · Noise setting: CSS parity-check decoding with calibrated soft information

For this code: Belief propagation supplies qubit reliabilities and ordered-statistics post-processing repairs non-convergence on the lattice's short cycles. It is a portable qLDPC baseline rather than a geometry-specialized decoder.

References: roffe2020decoding

Belief matching belief-matching standard

Uses belief propagation to infer correlated circuit faults and reweight a detector graph before minimum-weight matching chooses a correction.

Type: belief-matching · Noise setting: correlated circuit-level detector faults

For this code: Belief propagation first assigns correlated fault mechanisms to detector events, then reweights a matching graph. This retains the mature matching backend while correcting the independent-edge approximation.

References: higgott2023beliefmatching