# Color code[1]

## Description

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 [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].

## Rate

For general 2D manifolds, \(kd^2 \leq c(\log k)^2 n\) for some constant \(c\) [4], meaning that color codes with finite rate can only achieve an asymptotic minimum distance that is logarithmic in \(n\).

## Transversal Gates

Transversal CNOT can be implemented via braiding [5]. Universal transversal gates can be achieved in 3D color code using gauge fixing [6], lattice surgery [7], or code deformation [8][5].

## Gates

Magic-state distillation protocols [2].Non-clifford gates can be implemented via code switching [2].Lattice surgery scheme for 2D layout yields lower resource overhead when compared to analogous surface code scheme [9].

## Decoding

Projection decoder [2].Matching decoder gives low logical failure rate [10].Integer-program-based decoder [11].Restriction decoder [12].Cellular-automaton decoder for the \(XYZ\) color code [13].

## Fault Tolerance

Clifford gates can be performed fault-tolerantly on a suitable 2D lattice [1].Steane's ancilla-coupled measurement method [7]Gauge fixing can be used to switch between 2D and 3D color codes, thereby yielding fault-tolerant with constant time overhead using only local quantum operations [14].

## Code Capacity Threshold

\(12.6\%\) threshold for triangular color code with the restriction decoder [15].\(12.6\%\) threshold for triangular color code with the projection decoder ([16]) [17]\(8.7\%\) threshold for phase errors for the hexagonal color code with the projection decoder [16]\(\geq 6\%\) threshold with rescaling-based decoder [18].

## Threshold

\(\geq 6.25\%\) threshold for 2D color codes with error-free syndrome extraction, and \(0.1\%\) with faulty syndrome extraction [19].\(0.46\%\) for 3D codes with clustering decoder [20].\(1.9\%\) 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 [20].\(0.31\%\) noise threshold error rate for gauge code using clustering decoder [21].\(0.2\%\) with depolarizing circuit-level noise using two flag-qubits per stabilizer generator and the restriction decoder [15].\(0.143\%\) with depolarizing circuit-level noise using perfect-matching decoder [7].\(>0\%\) threshold with sweep decoder [2].

## Parents

- Calderbank-Shor-Steane (CSS) stabilizer code — See Ref. [2] for the chain complexes associated with the color code.
- Abelian topological code — When treated as ground states of the code Hamiltonian, 2D color code states on realize \(\mathbb{Z}_2\times\mathbb{Z}_2\) topological order [22], equivalent to the phase realized by two copies of the surface code [23].

## Children

- \([[15,1,3]]\) quantum Reed-Muller code — The \([[15,1,3]]\) code is a 3D color code.
- \([[7,1,3]]\) Steane code — Steane code is the smallest 2D color code.
- \([[8,3,2]]\) code — The \([[8,3,2]]\) code is the smallest non-trivial 3D color code.

## Cousins

- Kitaev surface code — Color code is equivalent to surface code in several ways [23][24]. 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.
- \(A_2\) hexagonal lattice code — The 2D color code is defined on a trivalent lattice such as the honeycomb lattice.
- 3D surface code — Color code is equivalent to surface code in several ways [23][24]. For example, the color code on a \(D\)-dimensional closed manifold is equivalent to multiple decoupled copies of the \(D-1\)-dimensional surface code.
- Floquet color code
- Galois-qudit topological code — Color code has been extended to Galois qudits.
- Generalized color code — The generalized color code for \(G=\mathbb{Z}_2\) reduces to the color 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 [25].
- Subsystem color code

## References

- [1]
- H. Bombin and M. A. Martin-Delgado, “Topological Quantum Distillation”, Physical Review Letters 97, (2006). DOI; quant-ph/0605138
- [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”. 1311.0277
- [4]
- N. Delfosse, “Tradeoffs for reliable quantum information storage in surface codes and color codes”, 2013 IEEE International Symposium on Information Theory (2013). DOI; 1301.6588
- [5]
- A. G. Fowler, “Two-dimensional color-code quantum computation”, Physical Review A 83, (2011). DOI; 0806.4827
- [6]
- H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”. 1311.0879
- [7]
- Andrew J. Landahl and Ciaran Ryan-Anderson, “Quantum computing by color-code lattice surgery”. 1407.5103
- [8]
- H. Bombin, “Clifford gates by code deformation”, New Journal of Physics 13, 043005 (2011). DOI
- [9]
- Felix Thomsen et al., “Low-overhead quantum computing with the color code”. 2201.07806
- [10]
- K. Sahay and B. J. Brown, “Decoder for the Triangular Color Code by Matching on a Möbius Strip”, PRX Quantum 3, (2022). DOI; 2108.11395
- [11]
- Ashley M. Stephens, “Efficient fault-tolerant decoding of topological color codes”. 1402.3037
- [12]
- C. Chamberland et al., “Triangular color codes on trivalent graphs with flag qubits”, New Journal of Physics 22, 023019 (2020). DOI; 1911.00355
- [13]
- Jonathan F. San Miguel, Dominic J. Williamson, and Benjamin J. Brown, “A cellular automaton decoder for a noise-bias tailored color code”. 2203.16534
- [14]
- H. Bombin, “Dimensional Jump in Quantum Error Correction”. 1412.5079
- [15]
- C. Chamberland et al., “Triangular color codes on trivalent graphs with flag qubits”, New Journal of Physics 22, 023019 (2020). DOI; 1911.00355
- [16]
- N. Delfosse, “Decoding color codes by projection onto surface codes”, Physical Review A 89, (2014). DOI; 1308.6207
- [17]
- N. Maskara, A. Kubica, and T. Jochym-O'Connor, “Advantages of versatile neural-network decoding for topological codes”, Physical Review A 99, (2019). DOI; 1802.08680
- [18]
- P. Parrado-Rodríguez, M. Rispler, and M. Müller, “Rescaling decoder for two-dimensional topological quantum color codes on 4.8.8 lattices”, Physical Review A 106, (2022). DOI; 2112.09584
- [19]
- D. S. Wang et al., “Graphical algorithms and threshold error rates for the 2d colour code”. 0907.1708
- [20]
- A. Kubica et al., “Three-Dimensional Color Code Thresholds via Statistical-Mechanical Mapping”, Physical Review Letters 120, (2018). DOI; 1708.07131
- [21]
- B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016). DOI; 1503.08217
- [22]
- M. Kargarian, H. Bombin, and M. A. Martin-Delgado, “Topological color codes and two-body quantum lattice Hamiltonians”, New Journal of Physics 12, 025018 (2010). DOI; 0906.4127
- [23]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015). DOI; 1503.02065
- [24]
- Arun B. Aloshious, Arjun Nitin Bhagoji, and Pradeep Kiran Sarvepalli, “On the Local Equivalence of 2D Color Codes and Surface Codes with Applications”. 1804.00866
- [25]
- H. Bombin et al., “Self-Correcting Quantum Computers”. 0907.5228

## Page edit log

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

## Zoo code information

## Cite as:

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