← Bivariate bicycle code

Entry facet

Representation ladder

R1 checks through R5 cross-model views, with canonicality and provenance.

  1. R1

    Checks

    Sparse Pauli generators in symplectic form, including stabilizer and gauge rows.

    • symplectic-css-checks derived by bivariate_bicycle.build -> CheckMatrix.from_css v1

      Expanding every polynomial monomial as a torus translation produces two sparse binary CSS matrices. Stacking their X and Z rows in symplectic form is the universal executable baseline used for rank, logical, and distance computations.

      References: bravyi2024bicycle

  2. R2

    Structure

    CSS pairs, chain complexes, lattices, and other structure behind the checks.

    • homological derived by bivariate_bicycle.build checkmatrix-to-CSS-complex v1

      The matrices form the three-term CSS complex R -> R^2 -> R with boundary maps [B;A] and [A|B]. Logical operators are middle homology classes, and the gross code's twelve logical qubits are the quotient dimension left after the two rank-66 boundary maps.

      References: lin2023twoblock, bravyi2024bicycle

    • toric-cylinder curated from sahay2026matching

      For decoding, selected symmetries reduce parts of the syndrome problem to toric-code-like cylinders on which matching applies. The reduction is an operational decoder dictionary, not a global phase equivalence.

      References: sahay2026matching

  3. R3

    Dynamics

    Ordered measurement schedules and their instantaneous stabilizer groups.

    No representation at this rung is recorded for this entry.

  4. R4

    Circuit & detectors

    Named syndrome-extraction circuits and detector error models.

    • tanner-circuit curated from bravyi2024bicycle

      Each weight-six check joins three qubits in each data layer. The Tanner incidence can be split into two planar degree-three coupling layers, turning sparse monomials into a concrete syndrome-extraction graph while keeping the schedule and detector model separate from the static code.

      References: bravyi2024bicycle, shaw2025morphing

  5. R5

    Cross-model views

    Executable or illuminating algebraic, ZX, fermionic, tensor, and statistical dictionaries.

    • bivariate-polynomial canonical

      A and B are sparse elements of F_2[x,y]/(x^l-1,y^m-1). Monomials are torus translations, transpose reverses a translation, and the two CSS check families are assembled without solving pairwise commutation constraints.

      References: bravyi2024bicycle

    • group-algebra curated from wang2023abelian

      The polynomial ring is the regular representation of the abelian group Z_l x Z_m. This view isolates the exact reason A and B commute and places BB codes inside two-block group-algebra codes without identifying the two families.

      References: wang2023abelian, lin2023twoblock

    • quotient-module curated from postema2025existence

      The common quotient R/(A,B) controls the number of encoded qubits. Polynomial factors and common roots explain when that quotient appears, why finite blocks can have high rate, and why the studied subclass is not an asymptotically good family.

      References: postema2025existence

    • anyon-koszul curated from chen2025anyon

      The same polynomial ideal can be read as excitation sectors, mobility constraints, and logical strings through a Koszul complex. This view exposes topological frustration and blocks the shortcut of declaring every finite BB code a trivial stack of independent toric codes.

      References: chen2025anyon