Fibonacci string-net code[1,2] 

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

When defined on a \(L \times L\) tailed honeycomb lattice on a torus, the code distance for ground-state encoding is \(L\).

Transversal Gates

A universal transversal gate set could be implemented in a folded version of this code using the techniques introduced in Ref [3].

Gates

Universal gate set for the ground-state encoding is implemented through topological operations corresponding to elements of the mapping class group, which is generated by Dehn-twists along non-contractible cycles. These Dehn-twists can be implemented using constant-depth circuits when allowing long-range permutations of qubits [4,5]. The mapping class group of a disk with \(m\) punctures is the braid group of \(m\) objects.Universal gate set for the fusion-space encoding is implemented through braiding of the computational anyons [2,6]. Circuit-based gates can be converted into braid patterns via quantum compiling algorithms [7,8].

Decoding

Clustering decoder (provides best known threshold for this code) [911].Fusion-aware iterative minimum-weight perfect matching decoder. Note that ordinary MWPM decoders do not produce a threshold with this code [11].Cellular automaton decoder [12].

Code Capacity Threshold

\(4.7\%\) for depolarizing noise, \(7.3\%\) for dephasing noise, and \(3.8\%\) for bit-flip noise with clustering decoder, assuming perfect measurements and gates [11]. See also Ref. [9].\(3.0\%\) for depolarizing noise, \(6.0\%\) for dephasing noise, and \(2.5\%\) for bit-flip noise with fusion-aware iterative MWPM decoder, assuming perfect measurements and gates [11].

Threshold

Between \(10^{-2}\%\) and \(5\cdot 10^{-2}\%\) for pair-creation and measurement noise [12].

Realizations

NMR: Implementation of braiding-based Hamamard gate on two qubits [13].Superconducting qubits: state preparation, fusion, and braiding [14].

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
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Zoo Code ID: fibonacci

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