## 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. This can equivalently constructed by braiding holes in a spherical geometry [2; Sec. 5].

## Protection

## Transversal Gates

## Gates

## Decoding

## Code Capacity Threshold

## Threshold

## Realizations

## Parent

## 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]
- G. Zhu, M. Hafezi, and M. Barkeshli, “Quantum origami: Transversal gates for quantum computation and measurement of topological order”, Physical Review Research 2, (2020) arXiv:1711.05752 DOI
- [4]
- 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
- [5]
- 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
- [6]
- M. Freedman, M. Larsen, and Z. Wang, “A modular functor which is universal for quantum computation”, (2000) arXiv:quant-ph/0001108
- [7]
- V. Kliuchnikov, A. Bocharov, and K. M. Svore, “Asymptotically Optimal Topological Quantum Compiling”, Physical Review Letters 112, (2014) arXiv:1310.4150 DOI
- [8]
- E. Génetay Johansen and T. Simula, “Fibonacci Anyons Versus Majorana Fermions: A Monte Carlo Approach to the Compilation of Braid Circuits in SU(2)k Anyon Models”, PRX Quantum 2, (2021) arXiv:2008.10790 DOI
- [9]
- S. Burton, C. G. Brell, and S. T. Flammia, “Classical simulation of quantum error correction in a Fibonacci anyon code”, Physical Review A 95, (2017) arXiv:1506.03815 DOI
- [10]
- 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
- [11]
- A. Schotte et al., “Quantum error correction thresholds for the universal Fibonacci Turaev-Viro code”, (2021) arXiv:2012.04610
- [12]
- A. Schotte, L. Burgelman, and G. Zhu, “Fault-tolerant error correction for a universal non-Abelian topological quantum computer at finite temperature”, (2022) arXiv:2301.00054
- [13]
- Y. Fan et al., “Experimental realization of a topologically protected Hadamard gate via braiding Fibonacci anyons”, (2022) arXiv:2210.12145
- [14]
- S. Xu et al., “Non-Abelian braiding of Fibonacci anyons with a superconducting processor”, (2024) arXiv:2404.00091

## Page edit log

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

## 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/edit/main/codes/quantum/categories/fibonacci.yml.