String-net code[1][2]

Description

Also called a Turaev-Viro or Levin-Wen model code. A family of topological codes, defined by a finite unitary spherical category \( \mathcal{C} \), whose generators are few-body operators acting on a cell decomposition dual to a triangulation of a two-dimensional surface (with a qudit of dimension \( |\mathcal{C}| \) located at each edge of the decomposition).

The codespace is the ground-state subspace of the Levin-Wen model Hamiltonian [1], a many-body Hamiltonian realizing the 3-manifold Turaev-Viro invariant [2][3]. Alternative constructions are possible, encoding information in the fusion space of the low-energy anyonic quasiparticle excitations of the model [4][2]. The fusion space can have dimension greater than one, allowing for topological quantum computation of logical information stored in the fusion outcomes.

Protection

Error-correcting properties established in Ref. [5].

Gates

Gates can be implemented through topological operations corresponding to elements of the mapping class group, which is generated by Dehn-twists along non-contractible cycles for triangulations of toroidal [6][7] and hyperbolic [8] manifolds. Whether or not a gate set is universal depends on the choice of input category; in some cases such as the Ising category, gates can be complemented by topological charge measurements to obtain a universal gate set.Alternatively, one could encode the logical quantum information into the degenerate fusion space of a number of computational anyons. In this case, a universal logical gate set can be implemented through the braiding of the computational anyons [4][9][2], e.g., for the case of the Fibonacci input category.

Decoding

Syndrome measurement circuits analyzed in Ref. [10].

Parents

Child

Cousin

  • Quantum-double code — String-net model reduces to the quantum-double model for group categories.

Zoo code information

Internal code ID: string_net

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: string_net

Cite as:
“String-net code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/string_net
BibTeX:
@incollection{eczoo_string_net, title={String-net code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/string_net} }
Permanent link:
https://errorcorrectionzoo.org/c/string_net

References

[1]
M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases”, Physical Review B 71, (2005). DOI; cond-mat/0404617
[2]
R. Koenig, G. Kuperberg, and B. W. Reichardt, “Quantum computation with Turaev–Viro codes”, Annals of Physics 325, 2707 (2010). DOI; 1002.2816
[3]
Alexander Kirillov Jr, “String-net model of Turaev-Viro invariants”. 1106.6033
[4]
Michael Freedman, Michael Larsen, and Zhenghan Wang, “A modular functor which is universal for quantum computation”. quant-ph/0001108
[5]
Y. Qiu and Z. Wang, “Ground subspaces of topological phases of matter as error correcting codes”, Annals of Physics 422, 168318 (2020). DOI; 2004.11982
[6]
G. Zhu, A. Lavasani, and M. Barkeshli, “Universal Logical Gates on Topologically Encoded Qubits via Constant-Depth Unitary Circuits”, Physical Review Letters 125, (2020). DOI; 1806.02358
[7]
G. Zhu, A. Lavasani, and M. Barkeshli, “Instantaneous braids and Dehn twists in topologically ordered states”, Physical Review B 102, (2020). DOI; 1806.06078
[8]
A. Lavasani, G. Zhu, and M. Barkeshli, “Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes”, Quantum 3, 180 (2019). DOI; 1901.11029
[9]
M. H. Freedman, M. J. Larsen, and Z. Wang, “The Two-Eigenvalue Problem and Density¶of Jones Representation of Braid Groups”, Communications in Mathematical Physics 228, 177 (2002). DOI; math/0103200
[10]
N. E. Bonesteel and D. P. DiVincenzo, “Quantum circuits for measuring Levin-Wen operators”, Physical Review B 86, (2012). DOI; 1206.6048

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/categories/string_net.yml.