Dihedral \(G=D_m\) quantum-double code[1,2] 

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

Universal topological quantum computation possible for certain groups such as \(G=D_3=S_3\) [5,6].

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, a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [24].
  • 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 [9], 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]
P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory”, (2024) arXiv:2405.11719
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Zoo Code ID: quantum_double_dihedral

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
BibTeX:
@incollection{eczoo_quantum_double_dihedral, title={Dihedral \(G=D_m\) quantum-double code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_double_dihedral} }
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“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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/groups/topological/quantum_double_dihedral.yml.