## Description

Quantum-double code whose codewords realize \(G=D_m\) topological order associated with a \(2m\)-element dihedral group \(D_m\). Includes the simplest non-Abelian order \(D_3 = S_3\) associated with the permutation group of three objects. The code can be realized as the ground-state subspace of the quantum double model, defined for \(D_m\)-valued qudits [1]. An alternative qubit-based formulation realizes the gauged \(G=\mathbb{Z}_3^2\) twisted quantum double phase [2], which is the same topological order as the \(G=D_4\) quantum double [3,4].

## Gates

## Realizations

Signatures of a phase equivalent to the \(G=D_4\) quantum double detected in a 27-qubit trapped-ion device by Quantinuum [7]. Preparation of ground states and braiding of anyons has also been performed. The phase was realized as a gauged \(G=\mathbb{Z}_3^2\) twisted quantum double [2], which is the same topological order as the \(G=D_4\) quantum double [3,4].

## Notes

See [8; Sec. 5.4] for an introduction to this code.The \( \Phi, \Lambda \) Decodoku game is based on the quantum double model for the group \(D_3=S_3\) of permutations on three letters.Popular summary of realization of non-Abelian topological order in Quanta Magazine.

## Parent

## Cousins

- Abelian TQD stabilizer code — Upon gauging some symmetries [9–11,11], a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [2–4].
- Cubic theory code — The ground-state subspace of the cubic theory model code in 2D reduces to that of the gauged \(G=\mathbb{Z}_3^2\) twisted quantum double model [12], realizing the same topological order as the \(G=D_4\) quantum double [3,4].

## References

- [1]
- A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003) arXiv:quant-ph/9707021 DOI
- [2]
- B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
- [3]
- M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
- [4]
- L. Lootens et al., “Mapping between Morita-equivalent string-net states with a constant depth quantum circuit”, Physical Review B 105, (2022) arXiv:2112.12757 DOI
- [5]
- C. Mochon, “Anyon computers with smaller groups”, Physical Review A 69, (2004) arXiv:quant-ph/0306063 DOI
- [6]
- G. K. Brennen, M. Aguado, and J. I. Cirac, “Simulations of quantum double models”, New Journal of Physics 11, 053009 (2009) arXiv:0901.1345 DOI
- [7]
- M. Iqbal et al., “Non-Abelian topological order and anyons on a trapped-ion processor”, Nature 626, 505 (2024) arXiv:2305.03766 DOI
- [8]
- J. K. Pachos, Introduction to Topological Quantum Computation (Cambridge University Press, 2012) DOI
- [9]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [10]
- 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
- [11]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [12]
- P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory”, (2024) arXiv:2405.11719

## Page edit log

- Victor V. Albert (2023-05-09) — most recent

## Cite as:

“Dihedral \(G=D_m\) quantum-double code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/quantum_double_dihedral