## Description

Member of a family of qubit CSS codes defined on particular \(D\)-dimensional graphs.

One family is defined on a \(D\)-dimensional graph which satisfies two properties: (1) the graph is a homogeneous simplicial \(D\)-complex obtained as a triangulation of the interior of a \(D\)-simplex, and (2) the graph is \(D+1\)-colorable. Qubits are placed on the \(D\)-simplices and generators are supported on suitable simplices [3–5]. Admissible graphs can be obtained via a fattening procedure [2]. See also a construction based on the more general quantum pin codes [6].

## Protection

As with the surface code, the code distance depends on the specific kind of lattice used to define the code. More precisely, the distance depends on the homology of logical string operators [3].

## Transversal Gates

Some color codes on \(D\)-dimensional lattices can transversally implement a gate at the \((D-1)\)st level of the Clifford hierarchy in the form of a \(Z\)-rotation by angle \(-\pi/2^D\) [4; Fig. 3].

## Decoding

In contrast to the surface code, the color code can suffer from unremovable hook errors due to the specifics of its syndrome extraction circuits. Fault-tolerant decoders thus have to utilize additional flag qubits.

## Fault Tolerance

The 6D color code is a self-correcting quantum memory and admits fault-tolerant universal gate set in 7D [7].

## Notes

See Ref. [3] for an overview of color codes.

## Parent

- Quantum pin code — Color codes are special cases of quantum pin codes [6; Sec. II.E]

## Children

- \([[2^r-1,1,3]]\) simplex code — Each \([[2^r-1,1,3]]\) simplex code is a color code defined on a simplex in \(r-1\) dimensions [8,9].
- 2D color code
- 3D color code
- Ball color code
- Hyperbolic color code
- Quasi-hyperbolic color code
- \([[16,6,4]]\) Tesseract color code

## Cousins

- Homological code — The color code on a \(D\)-dimensional closed manifold is equivalent to multiple decoupled copies of the \(D\)-dimensional surface code via a local constant-depth Clifford circuit [10–12]. This process can be viewed as an ungauging [13–15,15] of certain symmetries. Several hybrid color-surface codes exist [16,17].
- Self-correcting quantum code — The 6D color code is a self-correcting quantum memory and admits fault-tolerant universal gate set in 7D [7].
- Sarvepalli-Brown subsystem code — Sarvepalli-Brown subsystem codes can be derived from color codes [18; Thm. 3].
- Subsystem color code

## 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]
- H. Bombin, “An Introduction to Topological Quantum Codes”, (2013) arXiv:1311.0277
- [4]
- A. Kubica and M. E. Beverland, “Universal transversal gates with color codes: A simplified approach”, Physical Review A 91, (2015) arXiv:1410.0069 DOI
- [5]
- 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
- [6]
- C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
- [7]
- H. Bombin et al., “Self-Correcting Quantum Computers”, (2012) arXiv:0907.5228
- [8]
- H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
- [9]
- B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016) arXiv:1503.08217 DOI
- [10]
- 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
- [11]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
- [12]
- 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
- [13]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [14]
- 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
- [15]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [16]
- M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
- [17]
- N. Shutty and C. Chamberland, “Decoding Merged Color-Surface Codes and Finding Fault-Tolerant Clifford Circuits Using Solvers for Satisfiability Modulo Theories”, Physical Review Applied 18, (2022) arXiv:2201.12450 DOI
- [18]
- P. Sarvepalli and K. R. Brown, “Topological subsystem codes from graphs and hypergraphs”, Physical Review A 86, (2012) arXiv:1207.0479 DOI

## Page edit log

- Victor V. Albert (2023-11-13) — most recent
- Balint Pato (2023-01-11)
- Victor V. Albert (2022-01-05)
- Xiaozhen Fu (2021-12-12)

## Cite as:

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