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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].

Gates

Universal topological quantum computation is possible for certain groups such as \(G=D_3=S_3\) [3,4].\(U\)-model gate set [5], which can protect from circuit-level noise with the help of an anyon interferometer for the case of \(G=S_3\) [6].

Fault Tolerance

Universal topological quantum computation is possible for certain groups such as \(G=D_3=S_3\) [3,4].\(U\)-model gate set [5], which can protect from circuit-level noise with the help of an anyon interferometer for the case of \(G=S_3\) [6].

Code Capacity Threshold

Behavior under \(X\)-type noise (namely, diffusion of certain anyons) for the \(G=D_4\) case is related to the phase diagram of a disordered net model [7].

Notes

See [9][8; Sec. 5.4] for introductions 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.

Cousins

  • Abelian TQD code— A Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [2,10,11].
  • Symmetry-protected topological (SPT) code— The \(D_4\) quantum double model can be obtained by gauging [1221] symmetries of a Type III \(\mathbb{Z}_2^3\) SPT [2,22].
  • Hexagonal \(CZ\) code— The ground-state subspace of the hexagonal \(CZ\) code realizes the topological order of the \(G=\mathbb{Z}^3_2\) TQD model [2,23], which is the same topological order as the \(G=D_4\) quantum double [10,11].
  • Brickwork \(XS\) stabilizer code— The ground-state subspace of the brickwork \(XS\) stabilizer code realizes the topological order of the \(G=\mathbb{Z}^3_2\) TQD model [2,23], which is the same topological order as the \(G=D_4\) quantum double [10,11].
  • Modular-qudit surface code— The \(D_3\) quantum double model can be obtained by gauging [1221] the charge conjugation symmetry of the \(\mathbb{Z}_3\) surface code [22]. A magic-state preparation routine for the \(\mathbb{Z}_4\) surface code traverses through the \(D_4\) quantum double model [24].

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]
C. Mochon, “Anyon computers with smaller groups”, Physical Review A 69, (2004) arXiv:quant-ph/0306063 DOI
[4]
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
[5]
S. X. Cui, S.-M. Hong, and Z. Wang, “Universal quantum computation with weakly integral anyons”, Quantum Information Processing 14, 2687 (2015) arXiv:1401.7096 DOI
[6]
L. Chen, Y. Ren, R. Fan, and A. Jaffe, “A Universal Circuit Set Using the \(S_3\) Quantum Double”, (2025) arXiv:2411.09697
[7]
P. Sala, J. Alicea, and R. Verresen, “Decoherence and wavefunction deformation of \(D_4\) non-Abelian topological order”, (2024) arXiv:2409.12948
[8]
J. K. Pachos, Introduction to Topological Quantum Computation (Cambridge University Press, 2012) DOI
[9]
C. F. B. Lo, A. Lyons, R. Verresen, A. Vishwanath, and N. Tantivasadakarn, “Universal Quantum Computation with the \(S_3\) Quantum Double: A Pedagogical Exposition”, (2025) arXiv:2502.14974
[10]
M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
[11]
L. Lootens, B. Vancraeynest-De Cuiper, N. Schuch, and F. Verstraete, “Mapping between Morita-equivalent string-net states with a constant depth quantum circuit”, Physical Review B 105, (2022) arXiv:2112.12757 DOI
[12]
M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
[13]
J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, “Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems”, Physical Review X 5, (2015) arXiv:1407.1025 DOI
[14]
S. Vijay, J. Haah, and L. Fu, “Fracton topological order, generalized lattice gauge theory, and duality”, Physical Review B 94, (2016) arXiv:1603.04442 DOI
[15]
D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
[16]
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
[17]
A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
[18]
W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
[19]
K. Dolev, V. Calvera, S. S. Cree, and D. J. Williamson, “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
[20]
T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
[21]
D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
[22]
A. Lyons, C. F. B. Lo, N. Tantivasadakarn, A. Vishwanath, and R. Verresen, “Protocols for Creating Anyons and Defects via Gauging”, (2025) arXiv:2411.04181
[23]
P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory”, (2024) arXiv:2405.11719
[24]
S.-J. Huang and Y. Chen, “Generating logical magic states with the aid of non-Abelian topological order”, (2025) arXiv:2502.00998
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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.