## Description

Color code defined on a two-dimensional planar graph. Each face hosts two stabilizer generators, a Pauli-\(X\) and a Pauli-\(Z\) string acting on all the qubits of the face.

Most translation-invariant color codes are defined in trivalent planar graphs with three-colorable faces. The three admissible uniform tilings are the 6.6.6 (honeycomb) tiling, the 4.8.8 (square octagon) tiling, and the 4.6.12 tiling [3; Fig. 1]. More general admissible tilings can be obtained via a fattening procedure [2]; see also a construction based on the more general quantum pin codes [4].

Logical dimension is determined by the genus of the underlying surface (for closed surfaces) and the types of boundaries (for open surfaces). There are three types of boundaries corresponding to the three colors of the faces [5,6].

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. A string of one color must end in a boundary of that same color.

## Rate

## Transversal Gates

## Gates

## Decoding

## Parents

- Color code
- Generalized 2D color code — The generalized color code for \(G=\mathbb{Z}_2\) reduces to the 2D color code.
- Twist-defect color code — Twist-defect color codes reduce to 2D color codes when there are no defects. See Ref. [13] for an alternative non-CSS extension of 2D color codes.
- Modular-qudit color code — Modular-qudit 2D color codes reduce to 2D color codes for \(q=2\).
- Galois-qudit color code — Galois-qudit 2D color codes reduce to 2D color codes for \(q=2\).
- Abelian quantum-double stabilizer code — When treated as ground states of the code Hamiltonian, states of the color code on a torus geometry realize \(\mathbb{Z}_2\times\mathbb{Z}_2\) topological order [14], equivalent to the phase realized by two copies of the surface code via a local constant-depth Clifford circuit [15]. This process can be viewed as an ungauging [16–18,18] of certain symmetries.

## Children

- \([[4,2,2]]\) Four-qubit code — The \([[4,2,2]]\) code can be interpreted as a 2D color code on a square of the 4.8.8 or 4.6.12 tilings, or on a trapezoidal patch that makes up two-thirds of a hexagon of the 6.6.6 tiling [19,20]. Concatenating the \([[4,2,2]]\) code with the surface code is equivalent to removing stabilizer generators from the 4.8.8 color code [21]. Concatenating the \([[4,2,2]]\) code with two copies of the surface code yields the 4.6.12 color code [21]. A small 6.6.6 color code is a \([[4,1,2]]\) subcode with three weight-three stabilizer generators [6; Fig. 7]; this code is equivalent to a twist-defect surface code on a tetrahedron inscribed in a sphere [22] via a single-qubit Clifford circuit.
- Truncated trihexagonal (4.6.12) color code
- Square-octagon (4.8.8) color code
- Union-Jack color code
- Honeycomb (6.6.6) color code

## Cousins

- Kitaev surface code — The 2D color code is equivalent to multiple decoupled copies of the 2D surface code via a local constant-depth Clifford circuit [15,23,24]. This process can be viewed as an ungauging [16–18,18] of certain symmetries. 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 triangular directions to the stabilizer group and then taking the center of the resulting nonabelian group [19]. Both the surface and 2D color codes can be constructed from two distinct types of lattices, namely, 4-valent and 3-valent 3-colorable lattices, respectively [25].
- 3D 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 [26].
- Linear binary code — As CSS codes, variants of the 2D color code are constructed out of self-dual classical codes on cubic planar graphs [27].
- Hamiltonian-based code — 2D color code Hamiltonians can be simulated, with the help of perturbation theory, by two-dimensional weight-two (two-body) Hamiltonians with non-commuting terms [14].
- 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.
- Majorana color code — The Majorana color code is a Majorana stabilizer analogue of the 2D color code.
- Derby-Klassen (DK) code — The DK code on several tilings resembles the 2D color code with some vertex qubits removed [28].
- 2D subsystem color code
- Three-fermion (3F) subsystem code — The 2D color code is equivalent to two decoupled copies of the 3F code in the sense that the same anyon theory describes the low-energy excitations of both codes [29][6; Appx. B].

## References

- [1]
- H. Bombin and M. A. Martin-Delgado, “Topological Quantum Distillation”, Physical Review Letters 97, (2006) arXiv:quant-ph/0605138 DOI
- [2]
- 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
- [3]
- A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault-tolerant quantum computing with color codes”, (2011) arXiv:1108.5738
- [4]
- C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
- [5]
- H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
- [6]
- M. S. Kesselring et al., “The boundaries and twist defects of the color code and their applications to topological quantum computation”, Quantum 2, 101 (2018) arXiv:1806.02820 DOI
- [7]
- 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
- [8]
- 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
- [9]
- N. Delfosse, “Decoding color codes by projection onto surface codes”, Physical Review A 89, (2014) arXiv:1308.6207 DOI
- [10]
- A. M. Stephens, “Efficient fault-tolerant decoding of topological color codes”, (2014) arXiv:1402.3037
- [11]
- S.-H. Lee, A. Li, and S. D. Bartlett, “Color code decoder with improved scaling for correcting circuit-level noise”, (2024) arXiv:2404.07482
- [12]
- C. Gidney and C. Jones, “New circuits and an open source decoder for the color code”, (2023) arXiv:2312.08813
- [13]
- P. Padmanabhan et al., “Non-CSS color codes on 2D lattices : Models and Topological Properties”, (2022) arXiv:2112.13617
- [14]
- 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
- [15]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
- [16]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [17]
- 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
- [18]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [19]
- M. S. Kesselring et al., “Anyon Condensation and the Color Code”, PRX Quantum 5, (2024) arXiv:2212.00042 DOI
- [20]
- R. S. Gupta et al., “Encoding a magic state with beyond break-even fidelity”, Nature 625, 259 (2024) arXiv:2305.13581 DOI
- [21]
- B. Criger and B. Terhal, “Noise thresholds for the [4,2,2]-concatenated toric code”, Quantum Information and Computation 16, 1261 (2016) arXiv:1604.04062 DOI
- [22]
- S. Burton, E. Durso-Sabina, and N. C. Brown, “Genons, Double Covers and Fault-tolerant Clifford Gates”, (2024) arXiv:2406.09951
- [23]
- 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
- [24]
- 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
- [25]
- J. T. Anderson, “Homological Stabilizer Codes”, (2011) arXiv:1107.3502
- [26]
- H. Bombin, “Dimensional Jump in Quantum Error Correction”, (2016) arXiv:1412.5079
- [27]
- H. Oral, “Constructing self-dual codes using graphs”, Journal of Combinatorial Theory, Series B 52, 250 (1991) DOI
- [28]
- Derby, Charles. Compact fermion to qubit mappings for quantum simulation. Diss. UCL (University College London), 2023.
- [29]
- Zhenghan Wang. private communication, 2017.

## Page edit log

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

## Cite as:

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