← Eight-qubit color 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-stabilizer canonical

      Five independent Pauli rows—one all-X check and four Z checks—form the executable symplectic generator matrix. Rank and centralizer calculations leave three protected qubits and expose the distance-two logical classes.

      References: campbell2016smallest

  2. R2

    Structure

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

    • cube-incidence curated from campbell2016smallest

      Binary labels place the data qubits at cube vertices. The Z-check supports are coordinate faces, while the X-check touches every vertex.

      References: campbell2016smallest

  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.

    No representation at this rung is recorded for this entry.

  5. R5

    Cross-model views

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

    • transversal-gate curated from kubica2015universal

      The cited parity-overlap analysis turns a physical transversal diagonal pattern into logical CCZ, making the code an operations resource rather than only an error-detecting memory.

      References: campbell2016smallest, kubica2015universal