Lifted-product qLDPC codes

Lifted-product (LP) codes extend hypergraph products from field-valued matrices to free chain complexes over a group algebra or a more general ring. Each algebra element is expanded through a selected group representation into a sparse block matrix, allowing suitable lifts to reduce the product redundancy of ordinary HGP codes.

A typical code is written $LP(A,B)$ with entries of $A,B$ in $R=\mathbb F_q[G]$. Product boundary maps are formed over , after which the regular representation replaces each by a permutation matrix. Lifting before or after the ordinary product gives equivalent sparse block operators.

Protection depends strongly on the lift group, base matrices, and local expansion. A lift expands a small protograph into sparse checks but does not guarantee large distance for arbitrary inputs. Breakthrough families achieve almost-linear distance, and suitable later constructions attain constant rate and linear distance under explicit expansion or randomness assumptions.

The logical dimension of a general LP code has no simple formula based only on matrix sizes; it must be computed from the homology of the expanded boundary maps. Family-level constant-rate and almost-linear/linear-distance statements do not define a finite point without $A,B,G,q$, and the chosen representation.

LP structure can support logical operations based on dimension jump or code switching, but this is not a universal transversal property of every LP instance. A protocol must specify source and target codes, the lifted complex, switching measurements, recovery, and intermediate distance.

LP checks remain sparse CSS constraints and admit BP, BP+OSD, and specialized iterative decoders. Lift symmetries create structured short cycles, while noise bias and seed selection materially affect convergence. Finite bias-tailored comparisons must record the $X/Z$ prior, OSD order, and logical-failure definition.

Asymptotically good code parameters do not constitute a complete fault-tolerant circuit. Expanded checks may retain constant weight while inducing nonlocal hardware connections. Syndrome scheduling, ancillas, correlated faults, and real-time latency require finite-lift analysis; Atlas does not relabel code-capacity decoding results as circuit-level thresholds.

The official qLDPC software provides an LPCode constructor from groups, base matrices, and lifts and can connect the resulting CSS checks to BP-family decoders. Atlas has not yet selected an LP instance with public matrices, independently verified distance, and a QECirc circuit, so the topic exposes reproducibility requirements and an explicit unavailable state.

A trivial lift reduces to a Galois-qudit HGP code, while a free group action makes LP a specialization of balanced products. Fiber-bundle, Abelian LP, two-block group-algebra, and some bicycle constructions fit the LP framework. Quantum Tanner and expander LP codes are closely related but not definitionally identical.

A finite LP record must identify the coefficient field, group and representation, base-matrix sizes, lift order, expanded ranks and check weights, distance method, and decoding channel. The label “lifted product” plus an asymptotic theorem does not uniquely determine a computable instance.