Homological-product codes

For chain complexes $(C,\partial_C)$ and $(D,\partial_D)$, degree of the total complex is , with

A Künneth-type decomposition relates product logical classes to input homology, but shortest representatives may be altered by boundary cancellation. In the square homological-product setting, typical controls are $n=n_1n_2$, $k=k_1k_2$, , and . The distance upper bound is not an automatically attained equality.

The Bravyi–Hastings random asymptotic construction gives, with high probability,

$k=\Theta(n),\qquad d=\Theta(n),\qquad w=\Theta(\sqrt n).$

Homological maps can transfer logical states and Pauli-product measurements between encoded representations, enabling fault-tolerant gate constructions. The logical action is the induced map on homology, so [[n,k,d]] alone does not determine a gate; the chain map, ancilla block, measurement order, basis correspondence, and distance preservation must be specified.

Homological-product structure supports union-find-style decoding, while sparse CSS matrices can also be passed to BP+OSD. Evidence for the Atlas [[49,1,9]] point is a BP-OSD-10 word-error-rate curve under a depolarizing code-capacity model, not a syndrome-noise or circuit-level threshold.

A homological product by itself defines a static code. Three- and higher-dimensional products can provide syndrome redundancy for single-shot protocols, and representation switching can support universal fault-tolerant gates, but these results depend on dimension, complexes, and measurement gadgets. They do not automatically apply to the random [[49,1,9]] matrix.

Reproduction requires chain-group dimensions, every boundary matrix, grading convention, the selected total-complex segment, expanded $H_X,H_Z$, a logical homology basis, and decoder parameters. The published [[49,1,9]] benchmark is identified as a random HMP sample, without enough provenance to equate it to any of QECDB's four same-parameter self-dual CSS entries.

HGP is the homological product of two length-one chain complexes; higher-dimensional HGP uses additional factors. Balanced products further quotient by a shared group action, while fiber-bundle codes introduce twisting. These are construction-level lineage relations, not automatic finite-matrix or circuit equivalences.

For lookup, identify the degrees and lengths of the input complexes, the selected CSS three-term segment, random versus explicit construction, and strict qLDPC versus $w=\Theta(\sqrt n)$ low-weight families. At $(n,k)=(49,$, Code Tables gives general stabilizer bounds, not an HMP-family bound; same-parameter QECDB entries do not replace matrix provenance.