We show that given an instance of a bivariate bicycle (BB) code, it is possible to generate an infinite sequence of new BB codes using increasingly large covering graphs of the original code's Tanner graph. When a BB code has a Tanner graph that is a $h$-fold covering of the base BB code's Tanner graph, we refer to it as a $h$-\textit{cover code}. We show that for a BB code to be a $h$-cover code, its lattice parameters and defining polynomials must satisfy simple algebraic conditions relative to those of the base code. By extending the graph covering map to a chain map, we show there are induced projection and lifting maps on (co)homology that enable the projection and lifting of logical operators and, in certain cases, automorphisms between the base and the cover code. The search space of cover codes is considerably reduced compared to the full space of possible polynomials and we find that many interesting examples of BB codes, such as the $[[144,12,12]]$ gross code, can be viewed as cover codes. We also apply our method to search for BB codes with weight 8 checks and find many new codes, including a $[[64,14,8]]$ and $[[144,14,14]]$ code. For an $h$-cover code of an $[[n,k,d]]$ BB code with parameters $[[n_h = hn, k_h, d_h]]$, we prove that $k_h \geq k$ for any $h$, and $d_h \leq hd$ when $h$ is odd. Furthermore if $h$ is odd and $k_h = k$, we prove the distance lower bound $d \leq d_h$. We conjecture it is always true that an $h$-cover BB code of a base $[[n,k,d]]$ BB code obeys the distance bounds $d \leq d_h \leq hd$. While the focus of this work is on bivariate bicycle codes, we expect these methods to generalise readily to many group algebra codes and to certain code constructions involving hypergraph, lifted, and balanced products.
We show that given an instance of a bivariate bicycle (BB) code, it is possible to generate an infinite sequence of new BB codes using increasingly large covering graphs of the original code's Tanner graph. When a BB code has a Tanner graph that is a h-fold covering of the base BB code's Tanner graph, we refer to it as a h-\textit{cover code}. We show that for a BB code to be a h-cover code, its lattice parameters and defining polynomials must satisfy simple algebraic conditions relative to those of the base code. By extending the graph covering map to a chain map, we show there are induced projection and lifting maps on (co)homology that enable the projection and lifting of logical operators and, in certain cases, automorphisms between the base and the cover code. The search space of cover codes is considerably reduced compared to the full space of possible polynomials and we find that many interesting examples of BB codes, such as the [[144,12,12]] gross code, can be viewed as cover codes. We also apply our method to search for BB codes with weight 8 checks and find many new codes, including a [[64,14,8]] and [[144,14,14]] code. For an h-cover code of an [[n,k,d]] BB code with parameters [[nh=hn,kh,dh]], we prove that kh≥k for any h, and dh≤hd when h is odd. Furthermore if h is odd and kh=k, we prove the distance lower bound d≤dh. We conjecture it is always true that an h-cover BB code of a base [[n,k,d]] BB code obeys the distance bounds d≤dh≤hd. While the focus of this work is on bivariate bicycle codes, we expect these methods to generalise readily to many group algebra codes and to certain code constructions involving hypergraph, lifted, and balanced products.
Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-6 checks, exemplified by [[144,12,12]] with kd2/n=12. We develop the algebraic structure theory of BB-type codes with weight-8 checks (weight-4 generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula k=2dimR/(A,B) (forcing even k), a 4ℓm-element symmetry group on generator pairs, an X/Z distance equality dX=dZ, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At n=144 the pipeline returns a census of 53 codes whose strongest members surpass the BB benchmark: [[144,6,d≥15]] exceeds the benchmark distance 12 (certified d≥15), [[144,10,12]] reaches it with weight-8 checks, and [[144,16,10]] encodes a third more logical qubits at kd2/n=11.11 (7.4% below benchmark) while decoding no worse. At n=72, [[72,14,8]] attains kd2/n=12.44---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-8 codes at ≈0.1% pseudo-threshold versus ≈0.4% for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.
Bivariate bicycle (BB) quantum codes are a prominent finite-length family of quantum low-density parity-check codes, but their minimum distance is usually established numerically rather than read from the defining polynomials. We study the Z-logical quotient K/S over F2[x,y]/(xℓ−1,ym−1) and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity dZ=min(dA,dC). Using the Frobenius structure of the finite group algebra, we prove rA=rC=k/2 for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances 4,6,10,10,12,18 of the six standard BB codes of lengths 18 to 288 and enumerates every minimum-weight logical operator. The resulting census shows that [[108,8,10]] is the only one of the six whose distance is attained in a single component, with dC=10 and dA=12.
We study a weighted-shift formulation of bivariate bicycle group-algebra (BBGA) codes over finite group algebras. One binary shift acts by right multiplication and the other by left multiplication; associativity ensures that these shifts commute even for a nonabelian seed group. Bivariate polynomials in the shifts, including mixed monomials, therefore define CSS codes. We give explicit sparsity bounds and distinguish nonabelian seed groups from codes that are intrinsically inequivalent to abelian realizations. For D3, a pair of weighted permutation shifts generates a regular C6×C12 action and recovers the known [[144,12,12]] and [[144,14,14]] bivariate bicycle codes. We also present a computer-verified [[144,16,12]] code over A4×C6, with every check of weight eight. This example belongs to the established nonabelian two-block group-algebra family. Exhaustive low-weight enumeration and an intrinsic stabilizer-support invariant certify that it is inequivalent to any ordinary abelian BB or two-block group-algebra realization, even under qubit permutations combined with local Clifford operations. Finally, a nonuniform weighted-shift construction over F2[C3] yields a [[18,4,3]] code with check weights six and eight. An exact enumeration of its minimum stabilizer supports rules out every ordinary binary 2BGA realization at that length, under the same equivalence. This establishes that the unrestricted weighted-shift formulation extends beyond the ordinary 2BGA class, even with an abelian seed group.
In reference Bravyi et al., [Nature (London) 627, 778 (2024)NATUAS0028-083610.1038/s41586-024-07107-7], Bravyi et al. found examples of bivariate bicycle (BB) codes with similar logical performance to the surface code but with an improved encoding rate. In this work, we generalize a novel parity-check circuit design principle called morphing circuits and apply it to BB codes. We define a new family of BB codes whose parity check circuits require a qubit connectivity of degree five instead of six while maintaining their numerical performance. Logical input or output to an ancillary surface code is also possible in a biplanar layout. Finally, we develop a general framework for designing morphing circuits and present a sufficient condition for its applicability to two-block group algebra codes.
Quantum LDPC code discovery requires searching large algebraic design spaces while reliably certifying the parameters and equivalence classes of any candidates found. We introduce an LLM-guided evolutionary workflow in which language models mutate Python programs that generate bivariate-bicycle and perturbed bivariate-bicycle code ansätze. Across five campaigns, the system performed approximately 1{,}650 evolutionary iterations, screened about 2×105 candidate codes, and required ∼140 hours of computation and ∼US$400 in LLM inference cost. Candidate codes are evaluated through a staged validation pipeline combining GF(2) rank computation, distance estimation and certification, mixed-integer linear programming, BLISS Tanner-graph deduplication, decomposability analysis, and local-Clifford equivalence checks. At block length n≤360, the workflow identifies 465 distinct candidate codes: 97 CSS bivariate-bicycle codes and 368 non-CSS perturbed variants. The CSS search recovers known high-performing codes and finds new finite-length representatives, including an indecomposable [[288,16,12]] code and higher-weight codes with up to k=50 at distance d=8. The non-CSS search produces perturbed codes matching the gross-code figure of merit at [[144,12,12]], along with additional high-distance candidates reported as certified values or upper bounds according to MILP status. Overall, these results show that LLM-guided program evolution can serve as a practical tool for structured quantum-code discovery when paired with independent evaluation.
We construct three new weight-6 bivariate-bicycle (BB) codes—two [[336,20,12]] codes, inequivalent under the BB presentation symmetries, over Z₂ × Z₈₄ and a [[360,24,12]] code over Z₆ × Z₃₀—and certify exact distance 12 in both the X and Z sectors. They encode 25% and 50% more logical qubits than the best distance-12, weight-6 code reported by IBM's evolutionary search (k = 16) [Cruz-Benito et al., arXiv:2606.02418]. They are also the only ones: a symmetry-reduced enumeration of every connected weight-6 BB code with three-term polynomials, n ≤ 360 and k ≥ 20 eliminates 1,448 of the 1,584 candidates with checked logical operators of weight at most 11, and the remaining 136 presentations fall into three classes under group automorphisms, renormalisation and A ↔ B, represented by the new codes; within the polynomial ansatz of [Bravyi et al., Nature 627, 778 (2024)], every one of the 432 such candidates has distance at most 10. The same certificates settle the distances of the IBM ladder [Bravyi et al., Nature 627, 778 (2024)]—[[72,12,6]], [[90,8,10]], [[108,8,10]], the gross code [[144,12,12]] and the two-gross code [[288,12,18]], both sectors—and certify 19 ≤ d ≤ 24 for [[360,12]], whose distance we establish to be exactly 24 by exhaustive enumeration, resolving the upper bound reported in [Bravyi et al., Nature 627, 778 (2024)]. For the depth-8 syndrome-extraction circuit they certify circuit-level distance exactly 6 for [[72,12,6]] in both memory bases and at least 6 for every other code tested. Certified lower bounds are DRAT refutations for n ≤ 144 and replayable search trees above, whose single pruning lemma is proved in Lean 4; a reader replays any certificate with a standard-library Python verifier plus drat-trim or a 200-line C replayer, without the software that produced it. In a 1.1-million-shot circuit-level benchmark under one frozen BP-OSD decoder, the new codes' block failure probabilities at p = 0.004 are 3.6–6.2 times lower than the gross code's and 3.4–5.6 times higher than the two-gross code's, in both memory bases. Every computation—search, certification, replay and benchmark—ran on one laptop (Apple M3 Pro, 18 GB) with no GPU and no cluster. Contents of the release paper/ — the manuscript (LaTeX source, bibliography and PDF). certificates/ — one manifest per code and sector (<code_id>_<sector>.json), its standard-library Python verifier (verify_qec_<code_id>_<sector>.py), the proofs in proofs/ (DRAT proofs and gzipped proof trees), the problem files and C source of the tree replayer, the Lean 4 minimality lemma (qec_minimality.lean), the weight-6 envelope witnesses with their checkers, the distance table, FORMAT.md (the certificate format) and REPLAY.md (a clean replay of every certificate: 44 of 44 pass). simulation/ — the projected detector error models, the BP-OSD and BP-LSD evaluation scripts, every 2,000-shot chunk record with its seed, the decoder-tuning grid and the joint-model controls. SHA256SUMS — the SHA-256 of every file. Verification In certificates/: python3 verify_qec_<code_id>_<sector>.py rebuilds the code from its polynomials, checks the logical basis by rank, regenerates every CNF or problem file byte for byte and replays the proof; its last line is the verdict. DRAT rows need drat-trim (set DRAT_TRIM); tree rows need a C compiler. The Lean lemma checks with lake env lean qec_minimality.lean in a project with the pinned Lean 4 v4.33.1 and Mathlib 0df444a3 (exit status 0). Integrity: shasum -a 256 -c SHA256SUMS. Hardware Every computation — search, certificate production, replay and the benchmark — ran on one MacBook Pro (Apple M3 Pro, 11 CPU cores, 18 GB memory, macOS 15.5), with no GPU and no cluster. License: Licensed by SmartLedger Solutions under CC BY-NC 4.0 (LICENSE.md). Commercial use requires a written license from SmartLedger Solutions (labs@smartledger.solutions). The certificate producer is proprietary and not distributed; verification does not need it.