# 3D color code[1]

## Description

Color code defined on a four-valent four-colorable tiling of 3D space. Logical dimension is determined by the genus of the underlying surface (for closed surfaces) and types of boundaries (for open surfaces).

There are 101 different types of boundaries for any uniform tiling [2]; this was shown for the great rhombated cubic honeycomb (a.k.a. cantitruncated cubic honeycomb) uniform tiling, but is valid for general uniform tilings.

## Transversal Gates

Transversal action of \(T\) gates on color codes on general 3-manifolds realizes a \(CCZ\) gate on three logical qubits and is related to a topological invariant that is called the triple intersection number [3].Transversal \(S\) gate on color codes on general 3-manifolds corresponds to a higher-form symmetry [3].Universal transversal gates can be achieved using lattice surgery or code deformation [4,5].

## Gates

Magic-state distillation protocols [6].Non-clifford gates can be implemented via code switching [6].

## Decoding

Decoder that maps 3D color code to three copies of the 3D surface code [7].

## Parents

- Color code
- 3D lattice stabilizer code
- Modular-qudit color code — Modular-qudit 3D color codes reduce to 3D color codes for \(q=2\).
- Abelian topological code
- Triorthogonal code

## Children

- \([[8,3,2]]\) CSS code — The \([[8,3,2]]\) code is the smallest non-trivial 3D color code.
- Cubic honeycomb color code
- Tetrahedral color code

## Cousins

- 3D surface code — The 3D color code is equivalent to multiple decoupled copies of the 3D surface code via a local constant-depth Clifford circuit [8–10]. This process can be viewed as an ungauging [11–13,13] of certain symmetries. This mapping can also be done via code concatenation [14].
- Concatenated qubit code — The 3D color code is equivalent to multiple decoupled copies of the 3D surface code via a local constant-depth Clifford circuit [8–10]. This process can be viewed as an ungauging [11–13,13] of certain symmetries. This mapping can also be done via code concatenation [14].
- XS stabilizer code — The 3D color code on a particular lattice admits XS stabilizers; see talk by M. Kesselring at the 2020 FTQC conference.
- Symmetry-protected topological (SPT) code — Transversal action of \(T\) gates on color codes on general 3-manifolds realizes a \(CCZ\) gate on three logical qubits and is related to a topological invariant that is called the triple intersection number [3]. Transversal \(S\) gate on color codes on general 3-manifolds corresponds to a higher-form symmetry [3].
- Haah cubic code (CC) — The 3D color and cubic code families both include 3D codes that do not admit string-like operators.
- 2D color code — Gauge fixing can be used to switch between 2D and 3D color codes, thereby yielding fault-tolerant computation with constant time overhead using only local quantum operations [15].
- 3D subsystem color code

## References

- [1]
- 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
- [2]
- Z. Song and G. Zhu, “Magic Boundaries of 3D Color Codes”, (2024) arXiv:2404.05033
- [3]
- G. Zhu et al., “Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries”, (2023) arXiv:2310.16982
- [4]
- H. Bombin, “Clifford gates by code deformation”, New Journal of Physics 13, 043005 (2011) arXiv:1006.5260 DOI
- [5]
- A. G. Fowler, “Two-dimensional color-code quantum computation”, Physical Review A 83, (2011) arXiv:0806.4827 DOI
- [6]
- 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
- [7]
- A. B. Aloshious and P. K. Sarvepalli, “Projecting three-dimensional color codes onto three-dimensional toric codes”, Physical Review A 98, (2018) arXiv:1606.00960 DOI
- [8]
- 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
- [9]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
- [10]
- 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
- [11]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [12]
- L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
- [13]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [14]
- M. Vasmer and D. E. Browne, “Three-dimensional surface codes: Transversal gates and fault-tolerant architectures”, Physical Review A 100, (2019) arXiv:1801.04255 DOI
- [15]
- H. Bombin, “Dimensional Jump in Quantum Error Correction”, (2016) arXiv:1412.5079

## Page edit log

- Victor V. Albert (2023-11-13) — most recent
- Cella Kove (2023-06-20)

## Cite as:

“3D color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/3d_color