Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark

1 Introduction

• BB codes use three pure-monomial terms per generator, a restriction that enables thickness-two Tanner-graph embeddings; systematic exploration of weight-eight group-based codes remained open.Bravyi et al. noted that group-based codes with weight-eight checks could offer a better parameter trade-off.Source: §1 · p. 1
• At n=144, the 53-code census includes [[144, 6, d ≥15]] above the BB benchmark distance 12, [[144, 10, 12]] matching it, and [[144, 16, 10]] with 7.4% lower kd2/n.At n=72, [[72, 14, 8]] reaches kd2/n = 12.44, more than twice that of the same-length BB code, and decodes better.Source: §1 · p. 2
• The paper relaxes the BB restriction to three pure monomials, using four arbitrary bivariate monomials and an exactly validated search pipeline to construct and certify weight-eight-check codes.Its structural results include a dimension formula, generator-pair symmetry, equality of X- and Z-distances, and subgroup-coset distance bounds that give a design rule.Source: §1 · pp. 1–2

1.1 Related work

• Related constructions span group and coset-based codes, mirror codes, and other structural variants, while automated searches target BB and related qLDPC design spaces [4] [5] [6] [7] [8] [9] [10] [11] [12].The paper remains in the abelian CSS bivariate BB setting and emphasizes exact cross-validated distances, structural theory, and decoding and circuit-level benchmarks as complementary contributions.Source: §1.1 · pp. 2–3
• Prior distance and decoding work includes exact certification for univariate bicycle codes and LLM-guided BB-code discovery [32] [9], alongside matching and erasure decoding studies [13] [33].The paper identifies Imai’s theory of two-dimensional cyclic codes as the classical foundation for its dimension formula.Source: §1.1 · p. 3

2.1 CSS codes

• A CSS code is specified by binary matrices HX and HZ satisfying HX HZ^T = 0, equivalently each matrix’s row space lies in the other’s kernel.Its parameters are denoted [[n, k, d]], and a CSS code is degenerate when a stabilizer has weight below d.Source: §2.1 · p. 3

2.2 Bivariate bicycle codes

• The bivariate polynomial ring R = F2[x, y]/(x^ℓ − 1, y^m − 1) is identified with the group algebra of Zℓ × Zm.Each polynomial f has an ℓm × ℓm multiplication matrix Mf; multiplication of polynomials corresponds to multiplication of these matrices.Source: §2.2 · p. 3
• For A, B ∈ R, the bivariate bicycle code QC(A, B) has check matrices HX = [A | B] and HZ = [B^T | A^T], with code length n = 2ℓm.The transpose polynomial is fT(x, y) = f(x^−1, y^−1), and wt(f) counts the nonzero coefficients of f.Source: §2.2 · p. 3

3.1 The CSS condition is automatic

• Every pair A, B defines a CSS code because the CSS commutation condition holds automatically.The check matrices satisfy H_X H_Z^T = 0 in the commutative ring over characteristic 2.Source: §3.1 · p. 3

3.2 Exact dimension formula

• k = 2 dim R/(A, B), so the code dimension is always even.The check-matrix ranks equal dim(A, B); for odd ℓ,m, k can also be computed by counting common zeros of A and B on μ_ℓ × μ_m.Source: §3.2 · p. 4

3.3 Symmetries and search-space reduction

• The generator-pair symmetry group has order 4ℓm, and every orbit gives the same [[n, k, d]] code parameters.The X- and Z-distances are equal; the constant-term filter also retains every code while reducing the search space by a factor approaching ℓm.Source: §3.3 · p. 5

3.4 Regularity and degeneracy

• Weight-4 generators give an (8, 8)-regular Tanner graph with weight-8 stabilizer generators.If d > wt(A) + wt(B), the code is degenerate; thus every weight-4 pair with d ≥ 9 is degenerate.Source: §3.4 · p. 6

3.5 Subgroup coset operators: explicit kernel vectors and distance upper bounds

• Subgroup-coset indicators yield kernel vectors when A or B lies in the subgroup augmentation ideal, giving d ≤ |H| when such a vector is not a stabilizer.For coordinate subgroups, divisibility by (1 + x) or (1 + y) gives corresponding column- or row-vector bounds when those vectors are logical.Source: §3.5 · pp. 6–7
• Avoiding the augmentation ideals for every small subgroup is a design rule for high-distance constructions.In particular, avoid (1 + x)- and (1 + y)-divisibility, which otherwise gives d ≤ max(ℓ, m).Source: §3.5 · p. 7

3.6 Exact certification of the minimum distance

• Exact certification of d ≥ d0 reduces to checking that every kernel vector of weight below d0 lies in the corresponding stabilizer space.The criterion applies separately to X and Z distances.Source: §3.6 · p. 7

4.1 Pipeline

• The search samples normalized weight-w pairs, filters by k, then accepts only candidates meeting the requested distance bound with exact certification.The sampled search is used because exhaustive enumeration is infeasible; the refinement returns an exact distance at d0 or d0 + 1, or certifies d ≥ d0 + 2.Source: §4.1 · pp. 7–8

4.2 Exact distance verification by bit-mask DFS

• The bit-mask DFS exhausts weight-w kernel words and tests stabilizer membership, returning the exact distance through wmax or certifying a lower bound above it.A second Magma implementation was cross-validated on the census and independently checked the exhibited minimum-weight logical words.Source: §4.2 · p. 8

4.3 Translation-symmetry pruning

• Translation symmetry reduces each weight-w search to two anchored DFS runs, cutting the search tree by up to 2ℓm while preserving exactness.The reduction is complete because translation preserves weight, kernel membership, and stabilizer membership.Source: §4.3 · pp. 8–9

5.2 Empirical observations on the census

• Across the census, d≥10 occurs only for k≤16, while k≥18 forces d≤8; the (8,9) grid yields the d=12 and d≥15 codes.The census Pareto frontier is {[[144,6,≥15]], [[144,10,12]], [[144,16,10]], [[144,20,8]], [[144,24,6]]}; the (12,6) grid yields the largest dimensions, k≤24.Source: §5.2 · pp. 9–10

5.1 Setup and census

• At n=144, [[144,6,d≥15]] exceeds BB distance 12; [[144,10,12]] matches it, while [[144,16,10]] encodes four more qubits at kd²/n=11.11.The [[144,16,10]] code's kd²/n is 7.4% below the benchmark value 12.Source: §5.1 · p. 9, Table 1 · p. 10

5.3 Extension to n = 72

• At n=72, [[72,14,8]] reaches kd²/n=12.44, more than twice the BB benchmark value 6, and [[72,4,10]] exceeds the BB distance 6.Both reported distances are exact, unlike the BB distance upper bounds described in.Source: Table 2 · p. 10, §5.3 · pp. 10–11

5.4 Extended census over other grids and lengths

• The (9,8) grid yields [[144,8,9]], while the (10,10) grid gives n=200 codes including [[200,20,9]] with kd²/n=8.1.A broader grid scan is left for future work.Source: Table 3 · p. 11, §5.4 · p. 11

6 Decoding Performance

• The code-capacity benchmarks use ideal syndrome extraction and identical BP-OSD hyperparameters; with 5000 shots, the n = 144 pseudo-threshold gap is indicative rather than definitive.A full syndrome-extraction schedule is identified as the relevant next step for a hardware verdict.Source: §6 · pp. 11–12, §6.2 · p. 12
• 3.74% versus 3.29% per-qubit pseudo-threshold: the [[144, 16, 10]] code performs almost identically to the BB reference while encoding 33% more logical qubits.The comparison uses BP-OSD under code-capacity depolarizing noise.Source: §6 · p. 11, Figure 1 · p. 12

6.1 Decoding comparison at n = 72

• At p = 3%, the [[72, 14, 8]] code has pL = 1.62% versus 2.93% for the BB reference, and its logical error rate is lower throughout the sampled low-error regime.The pseudo-threshold interpolation is noise-sensitive and is not emphasized.Source: §6.1 · p. 12, Figure 2 · p. 13

6.2 Towards circuit-level fault tolerance

• ≈0.1% circuit-level pseudo-threshold for the weight-8 codes versus ≈0.4% and 0.2% for the weight-6 BB references, under the same six-round memory model.Weight-8 checks require depth 10 versus depth 7 for weight-6 checks, and the paper identifies reducing this cost as an open circuit question.Source: §6.2 · p. 13, Figure 3 · p. 14

• Four bivariate monomials per generator replace three pure monomials, enlarging the search space to weight-eight checks.Source: §7 · p. 14
• A validated bit-mask DFS computes distances exactly or certifies lower bounds, replacing the distance upper bounds reported for most codes in.Source: §7 · p. 14
• The paper proves the dimension formula and X/Z distance equality, adds symmetry and subgroup-coset bounds, and supplies certificates for independent re-derivation.Weight-8 Tanner graphs have thickness 3, compared with thickness 2 for weight-6 BB codes [4].Source: §7 · p. 14

• The exact dimension formula forces even k; the theory also establishes a 4ℓm-element symmetry group, equal X/Z distances, and subgroup-coset distance bounds that guide construction.Exact distance certification replaces estimates used in previous constructions.Source: §8 · p. 14
• The [[144, 6, d ≥15]] code exceeds the BB benchmark distance 12; [[144, 10, 12]] reaches that distance, while [[144, 16, 10]] encodes a third more logical qubits at kd2/n = 11.11.The [[144, 16, 10]] value is 7.4% below the benchmark kd2/n of 12 and decodes no worse.Source: §8 · p. 15
• At n = 72, [[72, 14, 8]] attains kd2/n = 12.44, more than twice the same-length BB code, and decodes measurably better.Source: §8 · p. 15
• The circuit-level memory experiment places the weight-8 codes at ≈0.1% pseudo-threshold versus ≈0.4% for the BB reference under an identical noise model.Source: §8 · p. 15

• All reported codes have explicit generator polynomials in Tables 1–3.Source: Data and Code Availability · p. 15
• For every exactly verified code, a machine-checkable certificate exhibits a minimum-weight logical operator independently re-checked in Magma.Source: Data and Code Availability · p. 15
• Supplementary materials include the full search–certify–verify pipeline, decoding scripts, and per-code check matrices for independent parameter re-derivation.Source: Data and Code Availability · p. 15