# Quantum-double code[1]

## Description

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, while local excitations are characterized by anyons. Different types of anyons are labeled by irreducible representations of the group's quantum double algebra, \(D(G)\) (a.k.a. Drinfield center) [3]. Not all isomorphic non-Abelian groups give rise to different quantum doubles [4].

For non-Abelian groups, alternative constructions are possible, encoding information in the fusion space of the low-energy anyonic quasiparticle excitations of the model [5–7]. 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.

Gapped boundaries of the models are classified by a subgroup \(K \subseteq G\) and a two-cocycle [3,8–10].

## Protection

## Encoding

## Gates

## Decoding

## Notes

## Parents

- Group GKP code — Quantum-double Hamiltonians can be expressed in terms of \(X\)- and \(Z\)-type operators of group-GKP codes; see [21; 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) [22].
- Hopf-algebra quantum-double code — Hopf-algebra quantum-double codes reduce to quantum-double codes when the Hopf algebra is a group algebra. Quantum-double codes for non-Abelian groups \(G\) are dual to Hopf-algebra quantum-double codes for Hopf algebras based on \(\text{Rep}(G)\) under the Tannaka-Krein duality [23][24; Fig. 1].

## Children

- \([[4,2,2]]_{G}\) four group-qudit code — The four group-qudit code is the smallest quantum double code.
- Dihedral \(G=D_m\) quantum-double code
- Abelian quantum-double stabilizer code — The anyon theory corresponding to (Abelian) quantum double codes is defined by an (Abelian) group.

## Cousins

- Hamiltonian-based code — Quantum double code Hamiltonians can be simulated, with the help of perturbation theory, by two-dimensional two-body Hamiltonians with non-commuting terms [25].
- Subsystem QECC — Subsystem versions of quantum-double codes have been formulated [26].
- Two-gauge theory code — Restricting 2-gauge theory constructions to a 2D manifold and replacing the 2-group with a group reproduces the phase of the Kitaev quantum double model [27].
- 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 [28].

## References

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- [2]
- S. X. Cui et al., “Kitaev’s quantum double model as an error correcting code”, Quantum 4, 331 (2020) arXiv:1908.02829 DOI
- [3]
- S. Beigi, P. W. Shor, and D. Whalen, “The Quantum Double Model with Boundary: Condensations and Symmetries”, Communications in Mathematical Physics 306, 663 (2011) arXiv:1006.5479 DOI
- [4]
- D. Naidu, “Categorical Morita equivalence for group-theoretical categories”, (2006) arXiv:math/0605530
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- R. Walter Ogburn and J. Preskill, “Topological Quantum Computation”, Quantum Computing and Quantum Communications 341 (1999) DOI
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- J. K. Pachos, Introduction to Topological Quantum Computation (Cambridge University Press, 2012) DOI
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- [18]
- S. Bravyi et al., “Adaptive constant-depth circuits for manipulating non-abelian anyons”, (2022) arXiv:2205.01933
- [19]
- 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
- [20]
- J. B. Kogut, “An introduction to lattice gauge theory and spin systems”, Reviews of Modern Physics 51, 659 (1979) DOI
- [21]
- V. V. Albert et al., “Spin chains, defects, and quantum wires for the quantum-double edge”, (2021) arXiv:2111.12096
- [22]
- M. Tomba et al., “Boundary algebras of the Kitaev Quantum Double model”, (2023) arXiv:2309.13440
- [23]
- O. Buerschaper and M. Aguado, “Mapping Kitaev’s quantum double lattice models to Levin and Wen’s string-net models”, Physical Review B 80, (2009) arXiv:0907.2670 DOI
- [24]
- O. Buerschaper et al., “Electric–magnetic duality of lattice systems with topological order”, Nuclear Physics B 876, 619 (2013) arXiv:1006.5823 DOI
- [25]
- C. G. Brell et al., “Toric codes and quantum doubles from two-body Hamiltonians”, New Journal of Physics 13, 053039 (2011) arXiv:1011.1942 DOI
- [26]
- P. Kumar, “A Class of Quantum Double Subsystem Codes”, (2011) DOI
- [27]
- A. Bullivant et al., “Topological phases from higher gauge symmetry in3+1dimensions”, Physical Review B 95, (2017) arXiv:1606.06639 DOI
- [28]
- 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

## Page edit log

- Victor V. Albert (2022-06-05) — most recent
- Victor V. Albert (2022-01-03)
- Ian Teixeira (2021-12-19)

## Cite as:

“Quantum-double code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_double