Quantum Tanner code

Linear distance follows jointly from expansion of the Cayley complex and robustness of the local tensor codes, not merely from low-weight checks. Suitable families achieve $K=\Theta(N)$ and $D=\Theta(N)$ with $O(1)$ check weight, under the explicit group-family and local-code hypotheses of the construction.

Physical qubits scale with the number of square faces, while fixed $A,B,C_A,C_B$ bound local check weight. Constant rate and linear distance describe a sequence with growing $|G|$, not a single tuple without a group and local codes, so Atlas keeps this record outside the finite-point landscape.

Good storage parameters do not specify a transversal gate set. Logical protocols must use the concrete homology, local-code symmetries, syndrome interface, and deformation path. Current mature guarantees center on storage, local decoding, and single-shot correction rather than a universal logical-gate template for all instances.

Sequential and parallel mismatch-decomposition decoders correct adversarial errors of linear weight under the required expansion and local-code robustness. The sequential algorithm runs in $O(N)$ time, while the parallel local-update algorithm achieves the ideal-syndrome guarantee in $O(\log N)$ rounds. This theorem is distinct from a stochastic logical-error benchmark.

Quantum Tanner codes admit single-shot decoding: one noisy round of constant-weight checks yields a correction whose residual is controlled by data and syndrome errors. Constant parallel iterations can maintain bounded residual noise across repeated rounds, followed by logarithmic iterations for final ideal readout. This adversarial-noise statement is not a gate-level depolarizing threshold.

The qLDPC package provides a QTCode constructor from $G,A,B,C_A,C_B$ for check generation and decoder experiments. Small explicit dihedral-group constructions are under study, but Atlas has not admitted a finite point with normalized matrices, independently verified distance, and a real syndrome circuit; QECDB and QECirc remain explicitly unavailable.

Quantum Tanner codes are a representative good-qLDPC construction connected to left-right Cayley-complex codes, classical Tanner codes, local tensor codes, and single-shot codes. They are closely related to expander lifted-product routes but must not be treated as synonyms in a database.

A reproducible record needs the group $G$, generators $A,B$, double-cover convention, local generator/check matrices, robustness assumptions, and expansion evidence. Without them, constant rate and linear distance remain conditional family-level statements and cannot be used to guess finite parameters or circuit artifacts.