Triangular color code[1] 


A planar color code defined on a trivalent lattice, typically the honeycomb or 4-8-8 (square octagon) lattice. Each boundary of the triangle intersects the lattice such that it only touches faces of two colors. The color of the boundary is then the other third color.

There are three types of boundaries corresponding to the three colors of the faces [2]. There are also three types of string operators, one for each color. A string of one color must end in a boundary of that same color.


Triangular color codes can be defined on the 4-8-8 lattice exist for any odd code distance \(d\). For any \(d\), \([[\frac{d^2-1}{2}+d, 1, d]]\) [3]. Triangular color codes can be defined on the honeycomb or 6-6-6 lattice for any odd code distance \(d\). For code distance \(d \geq 5\), the number of data qubits is \(\frac{(3d-1)^2}{4}\) [4].

Code Capacity Threshold

\(12.6\%\) threshold for triangular color code with the restriction decoder [4] and the projection decoder [5,6].\(8.7\%\) threshold for phase errors for the honeycomb triangular color code with the projection decoder [5].\(\geq 6\%\) threshold with rescaling-based decoder on the 4-8-8 triangular color code [7].\(44\%\) threshold under erasure noise for the 4-8-8 triangular color code [8] (see also [9]).



  • \([[4,2,2]]\) CSS code — \([[4,2,2]]\) code can be interpreted as a small rectangular color code on a trapezoidal patch of four qubits that makes up two-thirds of a hexagon [10,11].
  • \([[7,1,3]]\) Steane code — Steane code is the smallest triangular color code.



H. Bombin and M. A. Martin-Delgado, “Topological Quantum Distillation”, Physical Review Letters 97, (2006) arXiv:quant-ph/0605138 DOI
H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
A. J. Landahl and C. Ryan-Anderson, “Quantum computing by color-code lattice surgery”, (2014) arXiv:1407.5103
C. Chamberland et al., “Triangular color codes on trivalent graphs with flag qubits”, New Journal of Physics 22, 023019 (2020) arXiv:1911.00355 DOI
N. Delfosse, “Decoding color codes by projection onto surface codes”, Physical Review A 89, (2014) arXiv:1308.6207 DOI
N. Maskara, A. Kubica, and T. Jochym-O’Connor, “Advantages of versatile neural-network decoding for topological codes”, Physical Review A 99, (2019) arXiv:1802.08680 DOI
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) arXiv:2112.09584 DOI
H. M. Solanki and P. K. Sarvepalli, “Decoding Topological Subsystem Color Codes Over the Erasure Channel using Gauge Fixing”, (2022) arXiv:2111.14594
H. M. Solanki and P. Kiran Sarvepalli, “Correcting Erasures with Topological Subsystem Color Codes”, 2020 IEEE Information Theory Workshop (ITW) (2021) DOI
M. S. Kesselring et al., “Anyon condensation and the color code”, (2022) arXiv:2212.00042
R. S. Gupta et al., “Encoding a magic state with beyond break-even fidelity”, (2023) arXiv:2305.13581
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Zoo Code ID: triangular_color

Cite as:
“Triangular color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@incollection{eczoo_triangular_color, title={Triangular color code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Triangular color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.