A family of Abelian topological CSS stabilizer codes defined on a \(D\)-dimensional graph which satisfies two properties: The graph is (1) a homogeneous simplicial \(D\)-complex obtained as a triangulation of the interior of a \(D\)-simplex and (2) is \(D+1\)-colorable.
Qubits are placed on the \(D\)-simplices and generators are supported on suitable simplices . For 2-dimensional color code, the lattice must be such that it is 3-valent and has 3-colorable faces, such as a honeycomb lattice. The qubits are placed on the vertices and two stabilizer generators are placed on each face .
As with the surface code, the code distance depends on the specific kind of lattice used to define the code. More precisely, the distance depends on the homology of logical string operators .
In contrast to the surface code, the color code can suffer from unremovable hook errors due to the specifics of its syndrome extraction circuits. Fault-tolerant decoders thus have to utilize additional flag qubits.
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“Color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/color