Quantum-double code

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

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].

Encoding

For an \(L\times L\) lattice, deterministic state preparation can be done with a geometrically local unitary \(O(L)\)-depth circuit [7][8] or an \(O(\log{L})\)-depth unitary circuit with non-local two-qubit gates [9].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 [10] (see Ref. [11] for specific dihedral groups). Anyon-pair creation requires an adaptive circuit for any nonabelian \(G\) [10].

Decoding

For any solvable group \(G\), topological charge measurements can be done with an adaptive constant-depth circuit with geometrically local gates and measurements throughout [10].

Notes

The \( \Phi, \Lambda \) Decodoku game is based on the quantum double model for the group \(S_3\) of permutations on three letters.

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)

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.