# Fibonacci string-net code[1]

## Description

Quantum error correcting code associated with the Levin-Wen string-net model with the Fibonacci input category, admitting two types of encodings.

The first type of encoding is into the ground-state subspace of the Levin-Wen model Hamiltonian, defined on a cell decomposition (dual to a triangulation) of a manifold with a qubit on each link. The code space is the simultaneous \(+1\) eigenspace of a set of vertex operators and plaquette operators, which are defined by the fusion rules and the numerical data of the Fibonacci category, respectively. The degeneracy of the code space is \(4g\), were \(g\) is the genus of the surface on which the cell decomposition is defined.

The second type of encoding is into the degenerate fusion space of a number of anyonic quasiparticle excitations of the Levin-Wen model.

## Protection

## Transversal Gates

## Gates

## Decoding

## Threshold

## Parent

## Zoo code information

## 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]
- G. Zhu, M. Hafezi, and M. Barkeshli, “Quantum origami: Transversal gates for quantum computation and measurement of topological order”, Physical Review Research 2, (2020). DOI; 1711.05752
- [3]
- 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
- [4]
- G. Zhu, A. Lavasani, and M. Barkeshli, “Instantaneous braids and Dehn twists in topologically ordered states”, Physical Review B 102, (2020). DOI; 1806.06078
- [5]
- Michael Freedman, Michael Larsen, and Zhenghan Wang, “A modular functor which is universal for quantum computation”. quant-ph/0001108
- [6]
- R. Koenig, G. Kuperberg, and B. W. Reichardt, “Quantum computation with Turaev–Viro codes”, Annals of Physics 325, 2707 (2010). DOI; 1002.2816
- [7]
- Alexis Schotte et al., “Quantum error correction thresholds for the universal Fibonacci Turaev-Viro code”. 2012.04610

## Cite as:

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

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