Can Large Language Models Guide Discovery of Weight-Five Bivariate Bicycle Codes?

Weight five is a tighter polynomial family

Bivariate bicycle (BB) codes are a prominent family of Calderbank–Shor–Steane (CSS) quantum low-density parity-check codes. The most extensively studied BB codes use two trinomials, giving weight-six stabilizer generators. This paper extends an earlier LLM-guided program-evolution workflow to weight-five codes, where "weight five" denotes the Pauli weight of each stabilizer generator. For the regular binomial–trinomial (2+3) CSS constructions used here, the data-qubit degree also equals five, but the paper notes that circuit connectivity and gauge-check weight remain distinct quantities.

The search targeted bivariate codes with a 2+3 support split and block lengths 72≤n≤432. The CSS campaigns optimized kd²/n across complementary profiles of small and large periodic lattices, while the perturbed bivariate bicycle (PBB) campaigns prioritized non-CSS candidates with d≥5. The six CSS distance-evaluation lattices contain 13,656,867 normalized binomial–trinomial pairs, so the search space is finite but large.

The catalogue certifies exact distances for 227 classes

The four campaigns produced 13,923 positive-distance observations representing 1,297 distinct proposals. After excluding 155 proposals with an exact value or reported endpoint d≤2, the retained catalogue contains 12,036 observations across 1,142 proposals: 912 CSS and 230 PBB. Of these, 1,081 are nonbaseline proposals attributable to LLM-generated programs; the remaining 61 comprise 35 seed/null-only proposals, six without a retained attribution event, and 20 independently supplied fixed baselines.

Distance certification combined a targeted follow-up with a catalogue-wide low-weight search. The targeted follow-up established exact distances for 106 of 175 retained targets. The catalogue-wide search examined 492 CSS classes containing proposals without direct lower-bound evidence; it found two d=2 proposals that were excluded, resolved 62 classes and 92 proposals at d=3 or 4, and certified d≥5 in 428 classes and 764 proposals. Of the latter, 45 proposals also have explicit weight-five logical operators in both sectors and are therefore exact.

Merging distance evidence within each presentation class gives 227 exact classes and 403 classes with a certified lower bound and reported endpoint. The exact classes cover 436 proposals: 398 already had exact row-level status before the merge and 38 acquire it by class isomorphism. The paper reports that no class contains conflicting distances or incompatible bounds.

Connected CSS realizations reach [[180,4,14]]

Three exact disconnected parent codes have components that admit explicit shorter BB presentations. The [[384,16,10]] parent decomposes into four isomorphic components; the connected BB presentation on (ℓ,m)=(16,3) with A=1+y+x²y², B=1+x³ is BLISS-isomorphic to one component and therefore defines an exact [[96,4,10]] code. The three components of [[420,18,10]] admit the connected BB presentation on (ℓ,m)=(5,14) with A=1+y¹³+x⁴y¹¹, B=1+xy⁷, giving [[140,6,10]]. Their figures of merit (FOM = kd²/n) are 4.17 and 4.29.

Among the connected CSS codes in the catalogue, [[180,4,14]] has the largest exact FOM at 4.356; [[180,4,13]] provides another exact result with FOM 3.76. The paper also reports that the [[96,4,10]] realization is equivalent to a Lin–Pryadko archive construction over the cyclic group C₄₈ of order 48, adding an explicit bivariate realization and parent–component relation for that archive construction. Relative to the Voss [[96,4,8]] benchmark, the [[96,4,10]] parameter point has larger distance at the same (n,k).

Most large parent dimensions come from replication

A basis-independent connectivity analysis identifies 409 of the 1,142 retained proposals as disconnected: 307 of 912 CSS proposals (33.7%) and 102 of 230 PBB proposals (44.3%). At the class level, 268 of 630 classes are disconnected, including 166 of 227 exact classes (73.1%). For each disconnected proposal, the components have equal size, their dimensions sum to the parent k, and their colored Tanner graphs are pairwise isomorphic under BLISS canonical labelling.

Componentwise analysis of seven high-k CSS parents finds that each is a direct sum of pairwise-isomorphic components. All four retained CSS proposals with k≥40 are disconnected direct sums with exact distances d≤5. More generally, 408 of the 409 disconnected proposals have component dimension k₀∈{2,4,6,8}; the sole exception is [[420,20,5]]=2×[[210,10,5]]. No retained connected CSS proposal has k≥20: the largest connected dimension is 16, attained by the exact [[384,16,4]] code, while the largest connected PBB dimension is four. The paper concludes that most of the catalogue's large parent dimensions arise through replication.

LLM search yields more certified d≥5 codes

Among the 936 distinct CSS proposals from the LLM-guided campaign with a recorded positive distance, 816 (87.18%) are certified at d≥5. For comparison, three random-search controls each sample 6,444 CSS code proposals uniformly without replacement, using the same per-lattice and encoded-dimension sample counts as the LLM-guided campaign. In these controls, 4,672–4,785 proposals (72.50–74.26%) meet the same criterion.

The paper states that the LLM-guided campaign has the higher certified yield, while the random controls cover more connected classes, and every replicate samples the campaign-certified [[180,4,14]] class. The authors describe this as a weight-five transfer control that complements the broader uniform-random and genetic-algorithm benchmarks in their earlier work. The limitations note that the stratified-uniform experiment fixes the campaign's allocation by lattice and dimension band and tests selection within those strata, not allocation among them; its campaign denominator contains logged positive-distance observations, whereas the control denominator contains all completed selected evaluations, so the reported campaign fraction can exceed its attempted-call yield.

Cyclotomic kernels organize the connected classes

The exact connected CSS constructions share a common quotient-character structure. Writing the lattice translation group additively, let G=Z_ℓ×Z_m and let φ:G→Z_q be a surjective homomorphism. For a primitive qth root of unity ω in an extension field of F₂, a monomial xⁱyʲ is evaluated as ω^φ(i,j). If the two terms of the binomial have the same image and the three terms of the trinomial sum to zero, both BB polynomials vanish on this character, and its Frobenius orbit is a common-kernel sector.

The paper reports that a multiplicity-one order-three orbit has degree two and contributes four logical dimensions, while a multiplicity-one order-seven orbit has degree three and contributes six. Algebraic analysis organizes the connected CSS classes into order-3, order-7, and order-15 cyclotomic-kernel strata, and yields a 28-class taxonomy of two-direction presentations within the sampled lattice orders. The limitations note that this taxonomy classifies the displayed group-algebra presentations rather than all equivalent realizations.

PBB codes peak at [[216,4,10]]

For PBB codes, the strongest connected exact parameter point is [[216,4,10]], attained by two distinct component classes. A complete group-level calculation proves that none of the four [[12,2,3]] component classes is local-Clifford-equivalent to CSS. The shortest components with certified exact distances are a PBB [[12,2,3]] code and a CSS [[24,4,3]] code, both with FOM 1.50.

A post-search comparison certifies seven imported Lin–Pryadko archive constructions. For leading parameter triples also represented in that archive, the paper provides exact distance evidence, explicit bivariate presentations, and verified component reductions. The comparison also certifies the [[132,4,12]] construction over the nonabelian group C₁₁×S₃, whose FOM of 4.364 is the highest compiled exact weight-five value. The archive also reports constructions over nonabelian ambient groups with parameters [[168,4,14]] (FOM 4.67), [[180,4,15]] (5.00), and [[192,4,16]] (5.33), but their archived distances were obtained using a randomized routine; the paper's all-logical MILP attempts found matching weight-14, weight-15, and weight-16 witnesses and proved optimality for 1 of 8, 0 of 8, and 1 of 8 logical objectives within the per-logical time limit.

Connectivity should be tested during generation

The paper's structural analysis shows that 73.1% of the classes with exact distance certificates contain repeated connected components, and that most large parent dimensions arise through replication rather than through genuinely large connected codes. This matters for how readers compare candidates: a parent code with a large k may be a direct sum of small isomorphic components, and the component quotient reduces the 630 parent classes to 608 component classes (528 CSS and 80 PBB, with 206 exact and 402 nonexact classes).

The authors state that the component results motivate testing connectivity during generation and optimizing a canonical connected component directly in future campaigns. Across the investigated parameter range and current exact evidence, the CSS discoveries define the stronger finite-length envelope. The limitations note that the 630 presentation classes quotient symmetries of the stored generator graphs, not arbitrary stabilizer-basis changes or general local-Clifford operations, and are therefore an upper bound on code-equivalence classes; the PBB label "non-CSS" likewise refers to the tested transformation set, except for the four [[12,2,3]] component classes whose complete local-Clifford orbits were checked. The most decision-relevant priority is therefore to treat connectivity as a first-class search objective and to compare candidates at the connected-component level, where the exact [[180,4,14]] CSS point and the [[216,4,10]] PBB point define the strongest certified endpoints.

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