A family of abelian topological CSS stabilizer codes defined on a \(D\)-dimensional lattice which satisfies two properties: The lattice 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 .
For general 2D manifolds, \(kd^2 \leq c(\log k)^2 n\) for some constant \(c\) , meaning that color codes with finite rate can only achieve an asymptotic minimum distance that is logarithmic in \(n\).
Transversal CNOT can be implemented via braiding . Universal transversal gates can be achieved in 3D color code using gauge fixing , lattice surgery , or code deformation .
Magic-state distillation protocols .Non-clifford gates can be implemented via code switching .Lattice surgery scheme for 2D layout yields lower resource overhead when compared to analogous surface code scheme .
Projection decoder .Matching decoder gives low logical failure rate .Integer-program-based decoder .Restriction decoder .Cellular-automaton decoder for the \(XYZ\) color code .
Clifford gates can be performed fault-tolerantly on a suitable 2D lattice .Syndrome measurement .Steane's ancilla-coupled measurement method 
Code Capacity Threshold
\(\geq 6\%\) threshold with rescaling-based decoder .
\(\geq 6.25\%\) threshold for 2D color codes with error-free syndrome extraction, and \(0.1\%\) with faulty syndrome extraction .\(0.46\%\) for 3D codes with clustering decoder .\(1.9\%\) for 1D string-like logical operators and \(27.6\%\) for 2D sheet-like operators for 3D codes with noise models using optimal decoding and perfect measurements .\(0.31\%\) noise threshold error rate for gauge code using clustering decoder .\(0.143\%\) with depolarising circuit-level noise using perfect-matching decoder .\(>0\%\) threshold with sweep decoder .
- Kitaev surface code — Color code is equivalent to surface code in several ways . For example, the color code on a \(D\)-dimensional closed manifold is equivalent to multiple decoupled copies of the \(D-1\)-dimensional surface code.
- Triorthogonal code — The 3D color code is triorthogonal.
- 3D surface code — Color code is equivalent to surface code in several ways . For example, the color code on a \(D\)-dimensional closed manifold is equivalent to multiple decoupled copies of the \(D-1\)-dimensional surface code.
- Haah cubic code — The color and cubic code families both include 3D codes that do not admit string-like operators.
- Self-correcting quantum code — The 6D color code is a self-correcting quantum memory .
- Subsystem color code
Zoo code information
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“Color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/color