n
144
Paper: arXiv:2606.02418, "Evolutionary Discovery of Bivariate Bicycle Codes with LLM-Guided Search" (non-CSS PBB catalog, Campaign 5 therein; code_id 12_6_0193, bliss_hash 2176a91d34df3783).
Target: the general-stabilizer board's unrestricted x weight-8 (and looser) cells. A perturbed bivariate bicycle (PBB) code augments a CSS bivariate-bicycle code's two polynomials (A, B) with two more, (C, D), placed in the stabilizer matrix's block-1 z-part: `H = (A B | C D; 0 0 | B^T A^T) (A C^T + B D^T` symmetric over GF(2) is the only extra commutativity constraint this imposes). When C = D = 0 it degenerates to the CSS BB code; nonzero (C, D) produces genuinely mixed X/Z stabilizers, i.e. a general stabilizer code rather than a CSS one. This submission mines an already-completed, now-published non-CSS evolutionary campaign ("Campaign 5"/"campaign7") from a companion repository rather than running a fresh search here.
Source: github.com/qiskit-community/qcode-discovery @ d6abdbd, results/campaign7_report.md. An LLM-guided evolutionary search (OpenEvolve) over weight-5-ish 4-tuples (A, B, C, D) on ℓ=12, m=6 (and other lattices), with a 3-model ensemble (Claude Opus 4.6 34%, GPT-5.3-Codex 33%, Gemini 3.1 Pro 33%) proposing mutations over ~500 iterations. 18,588 raw candidate evaluations across 7 lattice dimensions (n=36 to n=360) were screened by k, isotropy (A C^T + B D^T symmetric), and a BP-OSD distance estimate; a two-stage dedup (tuple-level, then BLISS canonical-form hashing of the colored Tanner graph, evaluation/tanner_equivalence.py:canonical_hash_noncss) collapsed these to 368 structurally distinct non-CSS codes. Every surviving code was then screened for being secretly CSS up to a per-qubit Clifford (evaluation/clifford_equivalence.py: verify_not_lc_css — brute-force over 36 uniform single-qubit Clifford assignments plus a GF(2) affine solve for non-uniform {I,S}/{H,HS} patterns — and is_equivalently_css on the constructed code); this one is lc_any: false in the source repo's merged results (results/campaign7_publication_merged.jsonl, record `bliss_hash 2176a91d34df3783, code_id 12_6_0193`), i.e. genuinely non-CSS under every tested Clifford class.
(ℓ, m) = (12, 6), A = x + x^2y, B = 1 + xy^5 + x^{11}y^4, perturbation C = xy^3 + x^{10}y^3, D = xy^5 + x^{10}y^5 (0-indexed exponent pairs: A=[[1,2],[10,3],[10,4]], B=[[0,0],[1,5],[11,4]], C=[[1,3],[10,3]], D=[[1,5],[10,5]]). n = 2·12·6 = 144, k = 12 (GF(2) rank), max check weight 8.
results/campaign7_publication_merged.jsonl):BP-OSD estimate d_bposd = 12; a deep per-logical MILP pass (evaluation/distance_milp.py, symplectic formulation) then proved **all 24 logicals optimal** at weight 12 (logicals_optimal: 24/24, trust_level: "EXACT", wall time ~12,917s) — an exact distance, not an incumbent.
repository's own trusted tools (not the source claim): rebuilt the symplectic check matrix S = (A|B|C|D)-derived stabilizer rows via evaluation/pbb_code.py:build_pbb_code, then ran this repo's own cli/qldpc.py submit --dry-run (RIS search, 50,000 trials, plus the gf2_fast accelerator on the symplectic doubling at 2,000,000 trials). Independently found d<=12, matching the source's exact claim exactly — the two distance pipelines (deep MILP vs. randomized RIS/accelerator search) agree.
Final claim here (as this board records it): witness-backed upper_bound (every stabilizer-board distance is d<= until a Hamming-weight-per-side certifier can be adapted to mixed-stabilizer codes, per CONTRIBUTING.md); the source repository's own independent evidence tier is exact.
Not applicable — this submission mined an existing, already-vetted and now-published campaign result rather than running a new search in this repository.
Model (source campaign, per-code attribution unavailable at that stage of the project): ensemble of Claude Opus 4.6, GPT-5.3-Codex, and Gemini 3.1 Pro (34/33/33%), matches provenance.model. This submission's rebuild, independent re-verification, and packaging in *this* repository was done separately, using only this repository's own tooling (cli/qldpc.py submit, verify/).
from evaluation.pbb_code import build_pbb_code
code = build_pbb_code(ell=12, m=6,
A_terms=[(1,2),(10,3),(10,4)],
B_terms=[(0,0),(1,5),(11,4)],
C_terms=[(1,3),(10,3)],
D_terms=[(1,5),(10,5)])
# code.matrix is the (144, 288) symplectic stabilizer matrix [X-part | Z-part]
Source: github.com/qiskit-community/qcode-discovery @ d6abdbd, evaluation/pbb_code.py, results/campaign7_publication_merged.jsonl.