## Description

Also called a Turaev-Viro or Levin-Wen model code. A family of topological codes, defined by a finite unitary fusion 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.

String-net codes can equivalently be formulated in terms of unitary quantum groupoids [5].

## Protection

## Encoding

## Gates

## Decoding

## Parents

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

## Child

## Cousins

- Quantum-double code — String-net model reduces to the quantum-double model for group categories.
- 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 \(\mathbb{Z}_q\).

## References

- [1]
- M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases”, Physical Review B 71, (2005) arXiv:cond-mat/0404617 DOI
- [2]
- R. Koenig, G. Kuperberg, and B. W. Reichardt, “Quantum computation with Turaev–Viro codes”, Annals of Physics 325, 2707 (2010) arXiv:1002.2816 DOI
- [3]
- A. Kirillov Jr, “String-net model of Turaev-Viro invariants”, (2011) arXiv:1106.6033
- [4]
- M. Freedman, M. Larsen, and Z. Wang, “A modular functor which is universal for quantum computation”, (2000) arXiv:quant-ph/0001108
- [5]
- L. Chang, “Kitaev models based on unitary quantum groupoids”, Journal of Mathematical Physics 55, 041703 (2014) arXiv:1309.4181 DOI
- [6]
- Y. Qiu and Z. Wang, “Ground subspaces of topological phases of matter as error correcting codes”, Annals of Physics 422, 168318 (2020) arXiv:2004.11982 DOI
- [7]
- Y.-J. Liu et al., “Methods for Simulating String-Net States and Anyons on a Digital Quantum Computer”, PRX Quantum 3, (2022) arXiv:2110.02020 DOI
- [8]
- G. Zhu, A. Lavasani, and M. Barkeshli, “Universal Logical Gates on Topologically Encoded Qubits via Constant-Depth Unitary Circuits”, Physical Review Letters 125, (2020) arXiv:1806.02358 DOI
- [9]
- G. Zhu, A. Lavasani, and M. Barkeshli, “Instantaneous braids and Dehn twists in topologically ordered states”, Physical Review B 102, (2020) arXiv:1806.06078 DOI
- [10]
- A. Lavasani, G. Zhu, and M. Barkeshli, “Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes”, Quantum 3, 180 (2019) arXiv:1901.11029 DOI
- [11]
- 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) arXiv:math/0103200 DOI
- [12]
- D. Beckman et al., “Measurability of Wilson loop operators”, Physical Review D 65, (2002) arXiv:hep-th/0110205 DOI
- [13]
- N. E. Bonesteel and D. P. DiVincenzo, “Quantum circuits for measuring Levin-Wen operators”, Physical Review B 86, (2012) arXiv:1206.6048 DOI
- [14]
- G. Dauphinais and D. Poulin, “Fault-Tolerant Quantum Error Correction for non-Abelian Anyons”, Communications in Mathematical Physics 355, 519 (2017) arXiv:1607.02159 DOI

## Page edit log

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

## 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.