Color code

Color codes are CSS topological codes on suitably colorable cell complexes. A common two-dimensional construction places qubits on vertices of a trivalent, three-face-colorable lattice and assigns both $X$- and $Z$-type stabilizers to each face. Their geometry supports a richer transversal gate structure than ordinary surface-code patches.

Errors create syndromes with color structure on face checks. Logical operators are colored strings or string nets connecting compatible boundaries, and distance is the minimum weight of a nontrivial such object. Closed-manifold degeneracy follows topology; patches additionally depend on boundary colors and corners.

Color codes do not have one lattice-independent $[[n,k,d]]$ formula. Parameters depend on dimension, cellulation, topology, boundaries, and generator choice. Atlas includes the smallest triangular $[[7,1,3]]$ weight-four teaching instance; larger patches increase distance and circuit overhead.

Suitable two-dimensional triangular color codes support important Clifford operations transversally, including logical Hadamard and phase-type operations, with blockwise transversal CNOT. Higher-dimensional color codes can support higher levels of the Clifford hierarchy. The exact gate set depends on dimension, boundaries, and logical encoding.

Two-dimensional color-code syndromes can be projected or restricted to surface-code-like matching problems and then lifted back. Integer-programming, tensor-network, union-find, and learned methods also exist. Projection must preserve color and boundary constraints; circuit-level decoding includes propagated correlations and measurement faults.

Face-check circuits can propagate a single ancilla fault into a hook error. In some color-code schedules, reordering alone cannot remove every dangerous propagation, motivating flag ancillas, cat states, or verified measurement constructions. Effective distance must be assessed using circuit faults, not only the static stabilizer matrix.

Small color-code and equivalent Steane-code instances have been used to study logical states, transversal gates, and syndrome extraction. Scaling requires suitable connectivity, parallel schedules, leakage handling, and real-time decoding. The Atlas seven-qubit point describes a stabilizer equivalence class, not one device circuit.

The smallest triangular color code coincides with the Steane code. On closed manifolds, color codes are locally Clifford equivalent to multiple toric/surface-code copies, with boundaries requiring additional bookkeeping. Code deformation and gauge fixing can move between color, surface, and subsystem representations.

When comparing color-code results, check lattice, boundaries, check weights, noise level, and decoder. The $[[7,1,3]]$ parameters do not describe the whole family. Atlas binds a self-dual CSS QECDB record with automorphism-group order 168 as the finite Steane-equivalent instance and records that basis.