← 4.8.8 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-css-checks derived by color_488.build face incidence -> CheckMatrix.from_css v1

      The constructor expands every selected face boundary into one binary row and uses the same row space for X and Z checks. Their stacked symplectic generator matrix is the executable baseline for commutation, rank, and logical calculations.

      References: bombin2006topological, stephens2014decoding

  2. R2

    Structure

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

    • square-octagon-vertex-face-css canonical

      Red square faces have four vertices and blue/green bulk octagons have eight. Boundary truncation turns some octagons into four-vertex checks. The constructor makes the face-incidence matrix once and uses it for both HX and HZ, so commutation reduces to the even overlap of any two face boundaries.

      References: bombin2006topological, stephens2014decoding

    • boundary-logical-string derived by constructor boundary incidence and GF(2) row-space checks v1

      A connected same-Pauli string along any colored side commutes with every face because each incident check meets it evenly. On this three-boundary disc the three side representatives differ by stabilizers and encode one logical pair, rather than one logical qubit per side.

      References: stephens2014decoding, landahl2011color

  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.

    • restricted-dual-matching-graphs curated from stephens2014decoding

      Dualize the patch so colored face checks become vertices and data qubits become cells. Removing one color class at a time gives ordinary matching graphs; decoding solves those projections and then reconciles them into one color-code correction. The boundary nodes belong to the graph definition rather than an after-the-fact matching option.

      References: wang2010graphical, stephens2014decoding

    • random-three-body-ising curated from katzgraber2009threshold

      Under perfect-syndrome bit-flip noise, one data-qubit error couples the three colored checks meeting at that vertex. Maximum-likelihood decoding therefore maps to a disordered three-body spin model, making the three-way syndrome correlation explicit instead of replacing it by three independent pairings.

      References: katzgraber2009threshold, ohzeki2009accuracy

    • local-surface-code-unfolding curated from kubica2015unfolding

      A bounded-depth local Clifford transformation maps color-code degrees of freedom to surface-code sectors on derived lattices. This is an equivalence dictionary useful for decoding and phase comparison, not a claim that the square-octagon checks or triangular boundaries are literally surface-code checks.

      References: kubica2015unfolding, aloshious2018localequivalence