Quantum-double code

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

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

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].

Gates

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

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 [12].

Notes

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

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

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

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