# Color code[1]

## Description

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 [2]. 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 [3].

## Protection

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].

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.

## Gates

## Decoding

## Fault Tolerance

## Parent

## Children

## Cousins

- Generalized surface code — The color code on a \(D\)-dimensional closed manifold is equivalent to multiple decoupled copies of the \(D\)-dimensional surface code [10–12].
- Self-correcting quantum code — The 6D color code is a self-correcting quantum memory [13].
- Subsystem color code

## References

- [1]
- H. Bombin and M. A. Martin-Delgado, “Topological Quantum Distillation”, Physical Review Letters 97, (2006) arXiv:quant-ph/0605138 DOI
- [2]
- A. M. Kubica, The ABCs of the Color Code: A Study of Topological Quantum Codes as Toy Models for Fault-Tolerant Quantum Computation and Quantum Phases Of Matter, California Institute of Technology, 2018 DOI
- [3]
- H. Bombin, “An Introduction to Topological Quantum Codes”, (2013) arXiv:1311.0277
- [4]
- K. Sahay and B. J. Brown, “Decoder for the Triangular Color Code by Matching on a Möbius Strip”, PRX Quantum 3, (2022) arXiv:2108.11395 DOI
- [5]
- A. M. Stephens, “Efficient fault-tolerant decoding of topological color codes”, (2014) arXiv:1402.3037
- [6]
- C. Chamberland et al., “Triangular color codes on trivalent graphs with flag qubits”, New Journal of Physics 22, 023019 (2020) arXiv:1911.00355 DOI
- [7]
- J. F. S. Miguel, D. J. Williamson, and B. J. Brown, “A cellular automaton decoder for a noise-bias tailored color code”, Quantum 7, 940 (2023) arXiv:2203.16534 DOI
- [8]
- L. Berent et al., “Decoding quantum color codes with MaxSAT”, (2023) arXiv:2303.14237
- [9]
- A. J. Landahl and C. Ryan-Anderson, “Quantum computing by color-code lattice surgery”, (2014) arXiv:1407.5103
- [10]
- B. Yoshida, “Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes”, Annals of Physics 326, 15 (2011) arXiv:1007.4601 DOI
- [11]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
- [12]
- A. B. Aloshious, A. N. Bhagoji, and P. K. Sarvepalli, “On the Local Equivalence of 2D Color Codes and Surface Codes with Applications”, (2018) arXiv:1804.00866
- [13]
- H. Bombin et al., “Self-Correcting Quantum Computers”, (2012) arXiv:0907.5228

## Page edit log

- Victor V. Albert (2023-11-13) — most recent
- Balint Pato (2023-01-11)
- Victor V. Albert (2022-01-05)
- Xiaozhen Fu (2021-12-12)

## Cite as:

“Color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/color