← 6.6.6 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_666.build face incidence -> CheckMatrix.from_css v1

      Expanding the selected triangular face boundaries gives identical binary X- and Z-check matrices. Their sparse symplectic rows are the universal executable object used for commutation, rank, logical, and small-distance checks.

      References: bombin2006topological, chamberland2020triangular

  2. R2

    Structure

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

    • vertex-face-css canonical

      Remove every third point from an integer triangular index grid and read the removed points as face centers. The six surrounding sites form a bulk hexagon, while clipping at the three sides gives boundary faces. The resulting face-incidence matrix is used for both X and Z checks; even face weights and even pairwise overlaps make the CSS commutation visible before any rank calculation.

      References: bombin2006topological, chamberland2020triangular

    • colored-homology curated from bombin2007homological

      Red, green, and blue strings are deformable by multiplying face checks. A colored string may end only on the matching color boundary. In the triangular patch, representatives of different colors and Pauli types describe one protected logical pair rather than three independent qubits.

      References: bombin2007homological, kesselring2024anyon

  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.

    • folded-surface-code curated from kubica2015unfolding

      A local Clifford unfolding turns the triangular color-code data into attached surface-code sheets. The three color boundaries become a folded arrangement of rough and smooth boundary sectors, explaining why color-code strings branch while the unfolded strings look surface-code-like.

      References: kubica2015unfolding

    • random-three-body-ising curated from katzgraber2009threshold

      With perfect syndrome, maximum-likelihood decoding maps to a random three-body Ising problem: one data-qubit error couples the three colored checks meeting that vertex. The dictionary exposes correlations that a naive independent matching of defect pairs would discard.

      References: katzgraber2009threshold

    • anyon-condensation curated from kesselring2024anyon

      A boundary is specified by which anyons condense there, not merely by the pigment used in a drawing. This separates the three color boundaries used by this constructor from Pauli boundaries and gives a precise language for surgery interfaces and domain walls.

      References: kesselring2024anyon

    • locality-bound curated from bravyi2013classification

      The two-dimensional locality-preserving gate bound stops at the Clifford level. The color code saturates that useful level through its transversal Clifford structure, but universality needs a mechanism such as injection, distillation, switching, or surgery that is not one constant-depth local gate on the isolated patch.

      References: bravyi2013classification

    • tensor-network curated from farrelly2022localtensor

      Small color-code encoding tensors can be contracted and their logical operators pushed through the network. Contraction constructs a new larger code with its own boundary data; it is not permission to infer a finite 6.6.6 patch from the bulk tiling name alone.

      References: farrelly2022localtensor