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 data faults with perfect syndrome (code capacity)

For this code: Defects are paired on the periodic primal or dual lattice. The decoder minimizes one additive chain cost; it does not sum every degenerate path in a logical class, and its benchmark cannot be transferred to depolarizing, coherent, or faulty-syndrome models.

References: higgott2022pymatching, dennis2002topological, wang2010thresholds

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: repeated noisy data and syndrome records in a declared phenomenological or circuit detector model

For this code: Matching moves to spacetime. Round boundaries, measurement normalization, hook correlations, and the extraction schedule become part of the decoder identity, so no code-capacity number is reused here.

References: higgott2022pymatching, dennis2002topological, wang2003confinement, wang2010thresholds

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: periodic topological-code Pauli models with a specified syndrome history

For this code: Hierarchical coarse-graining retains approximate logical-class probabilities and trades accuracy for speed. Published behavior belongs to the chosen lattice, channel, scale rule, and syndrome model.

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: restricted independent channels exactly, and locally correlated Pauli channels approximately

For this code: Free-fermion or tensor contractions compare logical-class weights rather than selecting one light path. Exactness and runtime change with the channel and contraction width; they are not universal Toric properties.

References: bravyi2014efficient, chubb2021statistical

BP + ordered-statistics bp-osd research-grade

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: finite sparse CSS matrices with calibrated soft information and perfect syndrome

For this code: BP-OSD is a portable qLDPC baseline. Ordered-statistics post-processing can repair short-cycle message-passing failures, but no Toric-specific likelihood or threshold follows from the matrix alone.

References: roffe2020decoding

Minimum-weight perfect matching mwpm research-grade

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: detectable qubit loss with known erasure locations plus declared computational faults

For this code: Lost edges can be merged into superchecks and decoded using their known locations. The percolation boundary and the computational-error tradeoff are specific to detectable loss, not to unheralded leakage.

References: higgott2022pymatching, stace2009loss