# 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. For non-Abelian groups, alternative constructions are possible, encoding information in the fusion space of the low-energy anyonic quasiparticle excitations of the model [3–5]. 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.

## 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 [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].

## 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

- 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].

## References

- [1]
- A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003) arXiv:quant-ph/9707021 DOI
- [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]
- R. Walter Ogburn and J. Preskill, “Topological Quantum Computation”, Quantum Computing and Quantum Communications 341 (1999) DOI
- [4]
- C. Mochon, “Anyon computers with smaller groups”, Physical Review A 69, (2004) arXiv:quant-ph/0306063 DOI
- [5]
- J. K. Pachos, Introduction to Topological Quantum Computation (Cambridge University Press, 2012) DOI
- [6]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [7]
- M. Aguado and G. Vidal, “Entanglement Renormalization and Topological Order”, Physical Review Letters 100, (2008) arXiv:0712.0348 DOI
- [8]
- M. Aguado, “From entanglement renormalisation to the disentanglement of quantum double models”, Annals of Physics 326, 2444 (2011) arXiv:1101.0527 DOI
- [9]
- 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
- [10]
- R. König, B. W. Reichardt, and G. Vidal, “Exact entanglement renormalization for string-net models”, Physical Review B 79, (2009) arXiv:0806.4583 DOI
- [11]
- N. Tantivasadakarn et al., “Long-range entanglement from measuring symmetry-protected topological phases”, (2022) arXiv:2112.01519
- [12]
- S. Bravyi et al., “Adaptive constant-depth circuits for manipulating non-abelian anyons”, (2022) arXiv:2205.01933
- [13]
- 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
- [14]
- J. B. Kogut, “An introduction to lattice gauge theory and spin systems”, Reviews of Modern Physics 51, 659 (1979) DOI
- [15]
- V. V. Albert et al., “Spin chains, defects, and quantum wires for the quantum-double edge”, (2021) arXiv:2111.12096
- [16]
- M. Tomba et al., “Boundary algebras of the Kitaev Quantum Double model”, (2023) arXiv:2309.13440
- [17]
- 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