Balanced-product qLDPC codes

Balanced-product (BP) codes are CSS product codes built from two classical-code chain complexes carrying compatible actions of a common finite group $G$. The diagonal symmetry orbits of the ordinary tensor product are identified. This quotient can reduce the physical block length while preserving many logical degrees of freedom and nontrivial distance; it is a construction framework, not one fixed finite code.

For right and left $\mathbb F_2[G]$ chain complexes $C,D$, the balanced tensor product is

$C{\otimes }_{G}D=(C{\otimes }_{{\mathbb{F}}_2}D)\mathrm{/}⟨cg\otimes d-c\otimes$

Logical operators are homology and cohomology classes of the quotient total complex. Quotienting can change physical orbit count, logical dimension, and shortest nontrivial chains at different rates, so distance is not determined by the group order alone. The original explicit construction first produces an anisotropic horizontal subsystem family and then uses distance balancing to strengthen its weaker direction.

Seed automorphisms and group actions can induce physical permutations and logical Clifford transformations; generalized Dehn twists are available for selected cyclic balanced products. Such gates depend on the group, homology basis, and mapping class and are not a universal transversal gate set for every BP code.

The expanded CSS matrices can be decoded by BP, BP+OSD, or other sparse-graph methods, but short cycles, degeneracy, and orbit-induced correlations affect message quality. BP+OSD is an evidence-backed finite-length baseline rather than an intrinsic optimal BP decoder; single-shot claims additionally require syndrome redundancy and a stated measurement-noise model.

Bounded check weight constrains stabilizer support but does not by itself imply shallow, geometrically local, or fault-tolerant syndrome circuits. Modular mappings must account for intramodule and intermodule links, remote entanglement, ancillas, scheduling conflicts, hook errors, and effective circuit distance.

A reproducible object needs group generators and a multiplication or permutation representation, both seed boundaries, the group actions, quotient-orbit indexing, expanded sparse checks, a logical basis, and decoder settings. Atlas does not synthesize a representative finite instance from a family name or asymptotic theorem.

With a trivial group action, the construction reduces to an ordinary tensor or HGP code. Free actions connect it to lifted products, while cyclic actions overlap with fiber-bundle, quasi-cyclic, and selected 2BGA/BB descriptions. A construction-level reformulation is not proof that two finite database matrices are permutation- or local-Clifford equivalent.

For lookup, distinguish explicit theorems, high-probability results, and conjectures; determine whether $N$ is counted before or after quotienting; identify pre- or post-distance-balancing parameters; and distinguish gauge checks from stabilizers. Without group actions and seed matrices, only a family-level reference is justified.