Heavy-hex-native bivariate bicycle codes from evolutionary and LLM-guided search
Bivariate bicycle (BB) codes, introduced as a practical quantum LDPC family by Bravyi and co-workers, achieve approximately 12 physical qubits per logical qubit through their [[144, 12, 12]] “Gross” code, but published BB codes are not explicitly constrained to the heavy-hexagonal connectivity of current IBM superconducting processors. We report a systematic computational search for BB codes under the toroidal locality surrogate L_1^ 4 (a necessary condition for native heavy-hex embedding). Using random, evolutionary, and LLM-guided strategies we identify seven heavy-hex-compliant BB codes whose code-capacity figure of merit k d^2 / n upper-bound-exceeds the Gross code by factors 1.84–4.41; four lie on Smith-Normal-Form-distinct non-cyclic abelian-group tori not identified in the sources we checked, while three lie on cyclic tori and carry residual prior-art risk against generalized-bicycle tables we could not fully inspect. We provide an MIP-based upper bound d 24 for [[192, 8, 24]] (with asymmetric d_X = 30, d_Z = 24) and BP+OSD coset-decoding upper bounds for the remainder. Circuit-level memory experiments under a non-fault-tolerant syndrome-extraction schedule matched to a Gross reference reveal a stabilizer-weight-versus-pseudothreshold correlation consistent with heuristic scaling: across our four measured codes, break-even pseudothreshold drops from 5 10^-3 at stabilizer weight 6 (Gross reference) to 3 10^-4 at stabilizer weight 12 ([[196, 18, 24]]). Headline implication: our highest-S code is not deployable on current IBM hardware until fault-tolerant scheduling is developed for its weight-12 stabilizers. Evolutionary search clearly dominates random search in our single-seed comparison (95% upper-CI on this dominance is reported in-text). LLM-guided proposals, tested across five reasoning modes against a matched random control, achieve significantly higher valid-proposal rate (risk ratio 1.93, Fisher exact p = 1.2 10^-3) but do not extend the Pareto front ( 8% rule-of-three upper-CI on frontier-extension rate). The paper’s primary contribution is therefore a quantitative characterization of the weight-vs-pseudothreshold tradeoff; the codes themselves guide but do not solve the near-term-hardware problem.