现代 qLDPC 构造

This page is a navigation and comparison layer for modern qLDPC constructions, not a single check-matrix family. It organizes input objects, product/lift/quotient/twist mechanisms, proved guarantees, reproducible finite instances, and execution evidence. Definitions and theorems remain on the leaf topics.

Construction choice begins with the algebraic input: HGP uses classical checks; homological products use chain complexes; BP quotients complexes with a shared group action; LP uses matrices over rings or group algebras; fiber bundles add twisting; 2BGA reduces LP to one-by-one group-algebra inputs; GB and BB then select cyclic and two-factor Abelian groups.

Rate, distance, check weight, qubit degree, and syndrome redundancy are separate axes. The qLDPC label constrains sparsity only; constant rate, linear or almost-linear distance, single-shot syndrome protection, and effective circuit distance require additional theorems or circuit evidence.

Logical-gate capability is not an aggregate qLDPC property. Chain maps, group-automorphism permutations, fold-transversal Cliffords, code switching, logical-Pauli measurements, and gate teleportation all require a finite code, logical basis, ancilla blocks, measurement order, and fault-propagation analysis.

Decoder choice follows the hypotheses: local expander/Tanner algorithms need expansion and robustness; generic sparse CSS matrices often use BP+OSD as a finite-length baseline; GB/BB can exploit cyclic and syndrome-code structure. Erasure, biased, depolarizing, phenomenological, and circuit-level thresholds are not interchangeable.

Constant check weight does not automatically give a shallow or fault-tolerant syndrome circuit. Execution evidence must record ancillas, scheduling conflicts, connectivity, remote entanglement, hook errors, repeated measurement, real-time decoding, and effective circuit distance.

Ingestion begins by selecting the record level. Asymptotic families store hypotheses and scaling; finite instances store complete seeds, expanded matrices, and logical operators; execution artifacts add circuits, noise, and decoder configuration. A family label or [[n,k,d]] alone is insufficient for strict QECDB/QECirc identity.

The hierarchy follows construction containment: HGP specializes homological products; BP, LP, and fiber bundles admit reformulations under stated data; 2BGA is a one-by-one group-algebra LP and also has a BP formulation; GB is cyclic 2BGA; BB is two-factor Abelian 2BGA. Construction-level reformulation is not a finite permutation, scaling, or local-Clifford equivalence certificate.

Lookup order should be: record level; algebraic input and equivalence relation; theorem quantifiers; finite matrix identity; noise model and decoder; only then thresholds and resources. A database home page or broader parent-family filter remains unavailable rather than becoming a misleading link.