## 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

## Encoding

## Gates

## Decoding

## Parents

- Category-based quantum code
- Topological code — String-net codes can be realized using Levin-Wen model Hamiltonians, which realize various topological phases [1][2][3].

## Child

## Cousins

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

## 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]
- Yu-Jie Liu et al., “Methods for simulating string-net states and anyons on a digital quantum computer”. 2110.02020
- [7]
- 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
- [8]
- G. Zhu, A. Lavasani, and M. Barkeshli, “Instantaneous braids and Dehn twists in topologically ordered states”, Physical Review B 102, (2020). DOI; 1806.06078
- [9]
- 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
- [10]
- 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
- [11]
- N. E. Bonesteel and D. P. DiVincenzo, “Quantum circuits for measuring Levin-Wen operators”, Physical Review B 86, (2012). DOI; 1206.6048

## Page edit log

- Alexis Schotte (2022-01-24) — most recent
- David Aasen (2022-01-24)
- Victor V. Albert (2022-01-24)

## Zoo code information

## 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.