Three-dimensional color code[1] 


Three-dimensional version of the color code.

Transversal Gates

Universal transversal gates can be achieved using lattice surgery [2] or code deformation [3,4].


\(0.46\%\) for 3D codes with clustering decoder [5].\(1.9\%\) 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 [5].




  • 3D surface code — The 3D color code is equivalent to multiple decoupled copies of the 3D surface code [68].
  • Dynamical automorphism (DA) code — The parent topological phase of the 3D DA color code is realized by two copies of the 3D color code.
  • Two-dimensional color code — 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 [9].
  • Haah cubic code — The 3D color and cubic code families both include 3D codes that do not admit string-like operators.


H. Bombin and M. A. Martin-Delgado, “Exact topological quantum order inD=3and beyond: Branyons and brane-net condensates”, Physical Review B 75, (2007) arXiv:cond-mat/0607736 DOI
A. J. Landahl and C. Ryan-Anderson, “Quantum computing by color-code lattice surgery”, (2014) arXiv:1407.5103
H. Bombin, “Clifford gates by code deformation”, New Journal of Physics 13, 043005 (2011) arXiv:1006.5260 DOI
A. G. Fowler, “Two-dimensional color-code quantum computation”, Physical Review A 83, (2011) arXiv:0806.4827 DOI
A. Kubica et al., “Three-Dimensional Color Code Thresholds via Statistical-Mechanical Mapping”, Physical Review Letters 120, (2018) arXiv:1708.07131 DOI
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
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
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
H. Bombin, “Dimensional Jump in Quantum Error Correction”, (2016) arXiv:1412.5079
Page edit log

Your contribution is welcome!

on (edit & pull request)

edit on this site

Zoo Code ID: 3d_color

Cite as:
“Three-dimensional color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
@incollection{eczoo_3d_color, title={Three-dimensional color code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:

Cite as:

“Three-dimensional color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.