Color code[1,2] 


Member of a family of qubit CSS codes defined on particular \(D\)-dimensional graphs.

One family is defined on a \(D\)-dimensional graph which satisfies two properties: (1) the graph is a homogeneous simplicial \(D\)-complex obtained as a triangulation of the interior of a \(D\)-simplex, and (2) the graph is \(D+1\)-colorable. Qubits are placed on the \(D\)-simplices and generators are supported on suitable simplices [35]. Admissible graphs can be obtained via a fattening procedure [2]. See also a construction based on the more general quantum pin codes [6].


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 [3].

Transversal Gates

Some color codes on \(D\)-dimensional lattices can transversally implement a gate at the \((D-1)\)st level of the Clifford hierarchy in the form of a \(Z\)-rotation by angle \(-\pi/2^D\) [4; Fig. 3].


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.

Fault Tolerance

The 6D color code is a self-correcting quantum memory and admits fault-tolerant universal gate set in 7D [7].


See Ref. [3] for an overview of color codes.




  • Homological code — The color code on a \(D\)-dimensional closed manifold is equivalent to multiple decoupled copies of the \(D\)-dimensional surface code via a local constant-depth Clifford circuit [1012]. This process can be viewed as an ungauging [1315,15] of certain symmetries. Several hybrid color-surface codes exist [16,17].
  • Self-correcting quantum code — The 6D color code is a self-correcting quantum memory and admits fault-tolerant universal gate set in 7D [7].
  • Sarvepalli-Brown subsystem code — Sarvepalli-Brown subsystem codes can be derived from color codes [18; Thm. 3].
  • Subsystem color code


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Zoo Code ID: color

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“Color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@incollection{eczoo_color, title={Color code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.