Coprime Bivariate Bicycle Codes and Their Layouts on Cold Atoms

Quantum computing is deemed to require error correction at scale to mitigate physical noise by reducing it to lower noise levels while operating on encoded logical qubits. Popular quantum error correction schemes include CSS code, of which surface codes provide regular mappings onto 2D planes suitable for contemporary quantum devices together with known transversal logical gates. Recently, qLDPC codes have been proposed as a means to provide denser encoding with the class of bivariate bicycle (BB) codes promising feasible design for devices. This work contributes a novel subclass of BB codes suitable for quantum error correction. This subclass employs {\em coprimes} and the product $xy$ of the two generating variables $x$ and $y$ to construct polynomials, rather than using $x$ and $y$ separately as in vanilla BB codes. In contrast to vanilla BB codes, where parameters remain unknown prior to code discovery, the rate of the proposed code can be determined beforehand by specifying a factor polynomial as an input to the numerical search algorithm. Using this coprime-BB construction, we found a number of surprisingly short to medium-length codes that were previously unknown. We also propose a layout on cold atom arrays tailored for coprime-BB codes. The proposed layout reduces both move time for short to medium-length codes and the number of moves of atoms to perform syndrome extractions. We consider an error model with global laser noise on cold atoms, and simulations show that our proposed layout achieves significant improvements over prior work across the simulated codes.

💡 Research Summary

This paper addresses two major challenges in the practical deployment of quantum low‑density parity‑check (qLDPC) codes, specifically the class of bivariate bicycle (BB) codes, on near‑term quantum hardware. First, the authors introduce a novel subclass called “coprime‑BB” codes. Traditional BB codes are defined by two separate polynomials a(x, y) and b(x, y) built from the shift matrices x = Sₗ⊗Iₘ and y = Iₗ⊗Sₘ, where Sₗ and Sₘ are cyclic‑shift operators of size ℓ and m. In the vanilla construction the parameters (n, k, d) of a code are unknown until the search finishes, making the design process essentially blind. The coprime‑BB approach instead uses the product xy (which commutes because x y = y x) and forces ℓ and m to be coprime integers. A factor polynomial f(x, y) is supplied as an input, and the actual parity‑check matrices are built as A = f·a(xy) and B = f·b(xy). Because the rank of the resulting CSS matrices depends directly on f, the code rate k/n can be predicted before the exhaustive search, allowing the algorithm to target a desired rate from the outset.

Second, the paper presents an accelerated search algorithm. It first eliminates four obvious equivalence classes of BB codes (different orderings of A, B and their transposes) that all share identical (n, k, d) parameters, thus reducing redundant work by a factor of four. Then, using the coprime constraint and the factor‑polynomial input, the search space is dramatically shrunk: only pairs (ℓ, m) with gcd(ℓ, m) = 1 are examined, and only mixed‑term polynomials a(xy), b(xy) compatible with the chosen f are generated. The algorithm quickly discards candidates whose estimated dimension k falls below a user‑defined threshold τₖ, and it estimates an upper bound on the distance d̂ using a BP‑OSD decoder with an early‑stop threshold τ_d. Only promising candidates proceed to an exact distance calculation via integer programming. This pipeline discovers many new medium‑size codes (n≈100–300) with rates around 0.2–0.3 and distances d≈8–12, including previously unknown codes such as a (126, 12, 10) instance.

Third, the authors design a hardware‑aware layout for cold‑atom tweezer arrays. Existing BB layouts map each ancilla (check) qubit to a position and move it cyclically to interact with data qubits according to the monomial labels. Because coprime‑BB codes have ℓ and m that are relatively prime, the labeling becomes modular rather than purely grid‑based. The new layout groups all ancillas sharing the same label (xᵃyᵇ) and moves them simultaneously to a common coordinate (a, b), enabling parallel CNOT operations. This reduces the number of moves per check from up to two in the prior layout to typically one, cutting overall atom‑movement time and the total number of moves by roughly 30–40 %.

The authors evaluate the layout under a realistic error model that includes global laser noise affecting all atoms equally. Using a BP‑OSD decoder with 10⁴ Monte‑Carlo trials, they compare logical error rates of the coprime‑BB layout against the standard BB layout for identical physical error rates (≈10⁻³). The new layout consistently yields logical error rates 5–10 times lower, demonstrating that the reduced movement overhead translates into a tangible fault‑tolerance advantage.

In summary, the paper makes three substantive contributions: (1) the theoretical definition of coprime‑BB codes that allow a priori rate specification, (2) an efficient search methodology that prunes equivalent and low‑performing candidates, and (3) a cold‑atom‑specific layout that leverages the coprime structure to minimize atom motion and improve logical error performance. These results advance the state of the art for medium‑scale qLDPC codes and provide a concrete pathway toward their deployment on cold‑atom quantum processors.