Quantum-double code[1] 


Group-GKP stabilizer code whose codewords realize 2D modular gapped topological order defined by a finite group \(G\). The code's generators are few-body operators associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface (with a qudit of dimension \( |G| \) located at each edge of the tesselation).

The physical Hilbert space has dimension \( |G|^E \), where \( E \) is the number of edges in the tessellation. The dimension of the code space is the number of orbits of the conjugation action of \( G \) on \( \text{Hom}(\pi_1(\Sigma),G) \), the set of group homomorphisms from the fundamental group of the surface \( \Sigma \) into the finite group \( G \) [2]. When \( G \) is Abelian, the formula for the dimension simplifies to \( |G|^{2g} \), where \( g \) is the genus of the surface \( \Sigma \).

The codespace is the ground-state subspace of the quantum double model Hamiltonian. For non-Abelian groups, alternative constructions are possible, encoding information in the fusion space of the low-energy anyonic quasiparticle excitations of the model [35]. The fusion space of such non-Abelian anyons has dimension greater than one, allowing for topological quantum computation of logical information stored in the fusion outcomes.


Error-correcting properties established in Ref. [2]. The code distance is the number of edges in the shortest non contractible cycle in the tesselation or dual tesselation [6].


A depth-\(L^2\) circuit that grows the code out of a small patch on an \(L\times L\) square lattice using CMULT gates (i.e., "local moves") [7,8].For an \(L\times L\) lattice, deterministic state preparation can be done with a geometrically local unitary \(O(L)\)-depth circuit [8,9] or an \(O(\log{L})\)-depth unitary circuit with non-local two-qubit gates [7,10].For any solvable group \(G\), ground-state preparation and anyon-pair creation can be done with an adaptive constant-depth circuit with geometrically local gates and measurements throughout [11,12] (see Ref. [13] for specific dihedral groups). Anyon-pair creation requires an adaptive circuit for any non-Abelian \(G\) [12].


Universal topological quantum computation possible for certain groups [4,9].


For any solvable group \(G\), topological charge measurements can be done with an adaptive constant-depth circuit with geometrically local gates and measurements throughout [12].


See Ref. [14] for a review of gauge theory, which admits quantum-double topological phases.


  • Group GKP code — Quantum-double Hamiltonians can be expressed in terms of \(X\)- and \(Z\)-type operators of group-GKP codes; see [15; Sec. 3.3].
  • Twisted quantum double (TQD) code — The anyon theory corresponding to a quantum-double code is a TQD with trivial cocycle. These models realize local topological order (LTO) [16].



  • Generalized 2D color code — Generalized color code for group \(G\) on the 4.8.8 lattice is equivalent to a \(G\) quantum double model and another \(G/[G,G]\) quantum double model defined using the Abelianization of \(G\).
  • Kitaev surface code — A quantum-double model with \(G=\mathbb{Z}_2\) is the surface code. Non-stabilizer surface-code states can be prepared by augmenting the surface code with a quantum double model [17].


A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003) arXiv:quant-ph/9707021 DOI
S. X. Cui et al., “Kitaev’s quantum double model as an error correcting code”, Quantum 4, 331 (2020) arXiv:1908.02829 DOI
R. Walter Ogburn and J. Preskill, “Topological Quantum Computation”, Quantum Computing and Quantum Communications 341 (1999) DOI
C. Mochon, “Anyon computers with smaller groups”, Physical Review A 69, (2004) arXiv:quant-ph/0306063 DOI
J. K. Pachos, Introduction to Topological Quantum Computation (Cambridge University Press, 2012) DOI
E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
M. Aguado and G. Vidal, “Entanglement Renormalization and Topological Order”, Physical Review Letters 100, (2008) arXiv:0712.0348 DOI
M. Aguado, “From entanglement renormalisation to the disentanglement of quantum double models”, Annals of Physics 326, 2444 (2011) arXiv:1101.0527 DOI
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
R. König, B. W. Reichardt, and G. Vidal, “Exact entanglement renormalization for string-net models”, Physical Review B 79, (2009) arXiv:0806.4583 DOI
N. Tantivasadakarn et al., “Long-range entanglement from measuring symmetry-protected topological phases”, (2022) arXiv:2112.01519
S. Bravyi et al., “Adaptive constant-depth circuits for manipulating non-abelian anyons”, (2022) arXiv:2205.01933
R. Verresen, N. Tantivasadakarn, and A. Vishwanath, “Efficiently preparing Schrödinger’s cat, fractons and non-Abelian topological order in quantum devices”, (2022) arXiv:2112.03061
J. B. Kogut, “An introduction to lattice gauge theory and spin systems”, Reviews of Modern Physics 51, 659 (1979) DOI
V. V. Albert et al., “Spin chains, defects, and quantum wires for the quantum-double edge”, (2021) arXiv:2111.12096
M. Tomba et al., “Boundary algebras of the Kitaev Quantum Double model”, (2023) arXiv:2309.13440
K. Laubscher, D. Loss, and J. R. Wootton, “Universal quantum computation in the surface code using non-Abelian islands”, Physical Review A 100, (2019) arXiv:1811.06738 DOI
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Zoo Code ID: quantum_double

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“Quantum-double code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_double
@incollection{eczoo_quantum_double, title={Quantum-double code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_double} }
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“Quantum-double code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_double

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