# Quantum-double code[1]

## Description

A family of topological codes, defined by a finite group \( G \), whose 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 nonabelian groups, alternative constructions are possible, encoding information in the fusion space of the low-energy anyonic quasiparticle excitations of the model [3][4][5]. The fusion space of such nonabelian anyons has dimension greater than one, allowing for topological quantum computation of logical information stored in the fusion outcomes.

## Protection

## Encoding

## Decoding

## Notes

## Parents

- Group GKP code — Quantum-double models admit stabilizer-like \(X\)- and \(Z\)-type operators [12], and the codes can be formulated as group GKP codes.
- Topological code — Quantum-double code Hamiltonians admit topological phases associated with finite groups \(G\).

## Cousins

- Modular-qudit surface code — A quantum-double model with \(G=\mathbb{Z}_q\) is the qudit surface code.
- String-net code — String-net model reduces to the quantum-double model for group categories.
- Kitaev surface code — A quantum-double model with \(G=\mathbb{Z}_2\) is the surface code.

## References

- [1]
- A. Y. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003). DOI; quant-ph/9707021
- [2]
- S. X. Cui et al., “Kitaev's quantum double model as an error correcting code”, Quantum 4, 331 (2020). DOI; 1908.02829
- [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). DOI; quant-ph/0306063
- [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). DOI; quant-ph/0110143
- [7]
- G. K. Brennen, M. Aguado, and J. I. Cirac, “Simulations of quantum double models”, New Journal of Physics 11, 053009 (2009). DOI; 0901.1345
- [8]
- M. Aguado, “From entanglement renormalisation to the disentanglement of quantum double models”, Annals of Physics 326, 2444 (2011). DOI; 1101.0527
- [9]
- M. Aguado and G. Vidal, “Entanglement Renormalization and Topological Order”, Physical Review Letters 100, (2008). DOI; 0712.0348
- [10]
- Sergey Bravyi et al., “Adaptive constant-depth circuits for manipulating non-abelian anyons”. 2205.01933
- [11]
- Ruben Verresen, Nathanan Tantivasadakarn, and Ashvin Vishwanath, “Efficiently preparing Schrödinger's cat, fractons and non-Abelian topological order in quantum devices”. 2112.03061
- [12]
- Victor V. Albert et al., “Spin chains, defects, and quantum wires for the quantum-double edge”. 2111.12096

## Page edit log

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

## Zoo code information

## Cite as:

“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/tree/main/codes/quantum/groups/quantum_double.yml.