# Two-dimensional color code[1]

## Description

Two-dimensional version of the color code, defined on a two-dimensional trivalent planar graph with 3-colorable faces. Each face hosts two stabilizer generators, a Pauli-\(X\) and a Pauli-\(Z\) string acting on all the qubits of the face.

String operators are defined on the paths along the edges of the qubit. These paths can have branching points. Each path has two string operators, one corresponding to the \(X\) basis and one corresponding to the \(Z\) basis. In correspondence with the coloring of the lattice faces, string operators also come in three colors. String operators commute or anti-commute. They anti-commute when they cross an odd number of times and have a different color and type. String operators correspond to encoded \(X\) or \(Z\) operators when they are closed and not boundaries.

As CSS codes, variants of the 2D color code are constructed out of self-dual codes on cubic planar graphs [2].

## Rate

## Transversal Gates

## Gates

## Fault Tolerance

## Threshold

## Parents

- Color code
- Generalized 2D color code — The generalized color code for \(G=\mathbb{Z}_2\) reduces to the 2D color code.
- Abelian quantum double stabilizer code — When treated as ground states of the code Hamiltonian, states of the color code on a torus geometry on realize \(\mathbb{Z}_2\times\mathbb{Z}_2\) topological order [10], equivalent to the phase realized by two copies of the surface code [11].

## Child

## Cousins

- Galois-qudit topological code — The 2D color code has been extended to Galois qudits.
- Kitaev surface code — The 2D color code is equivalent to multiple decoupled copies of the 2D surface code [11–13]. Conversely, the 2D color code can condense to form the 2D surface code in nine different ways, i.e., by adding two body hopping terms along one of its three hexagonal directions to the stabilizer group and then taking the center of the resulting nonabelian group [14].
- Three-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 [15].
- Dynamical automorphism (DA) code — The parent topological phase of the 2D DA color code is realized by two copies of the 2D color code.
- Floquet color code — The parent topological phase of the Floquet color code is the \(\mathbb{Z}_2\times\mathbb{Z}_2\) 2D color-code phase.

## References

- [1]
- H. Bombin and M. A. Martin-Delgado, “Topological Quantum Distillation”, Physical Review Letters 97, (2006) arXiv:quant-ph/0605138 DOI
- [2]
- H. Oral, “Constructing self-dual codes using graphs”, Journal of Combinatorial Theory, Series B 52, 250 (1991) DOI
- [3]
- N. Delfosse, “Tradeoffs for reliable quantum information storage in surface codes and color codes”, 2013 IEEE International Symposium on Information Theory (2013) arXiv:1301.6588 DOI
- [4]
- A. G. Fowler, “Two-dimensional color-code quantum computation”, Physical Review A 83, (2011) arXiv:0806.4827 DOI
- [5]
- F. Thomsen et al., “Low-overhead quantum computing with the color code”, (2022) arXiv:2201.07806
- [6]
- D. S. Wang et al., “Graphical algorithms and threshold error rates for the 2d colour code”, (2009) arXiv:0907.1708
- [7]
- C. Chamberland et al., “Triangular color codes on trivalent graphs with flag qubits”, New Journal of Physics 22, 023019 (2020) arXiv:1911.00355 DOI
- [8]
- A. J. Landahl and C. Ryan-Anderson, “Quantum computing by color-code lattice surgery”, (2014) arXiv:1407.5103
- [9]
- 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
- [10]
- 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) arXiv:0906.4127 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]
- 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
- [13]
- 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
- [14]
- M. S. Kesselring et al., “Anyon condensation and the color code”, (2022) arXiv:2212.00042
- [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:

“Two-dimensional color code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/2d_color