## Description

Code whose logical subspace is labeled by different fusion outcomes of Ising anyons present in the Ising-anyon topological phase of the Kitaev honeycomb model [1]. Each logical qubit is constructed out of four Majorana operators, which admit braiding-based gates due to their non-Abelian statistics and which can be used for topological quantum computation. Ising anyons also exist in other phases, such as the fractional quantum Hall phase [4].

The Kitaev honeycomb model can be formulated as a qubit subsystem stabilizer code based on the \(\mathbb{Z}_2^{(1)}\) Abelian anyon theory, which is non-chiral and non-modular [5; Sec. 7.3]. The model realizes all anyon theories of the 16-fold way, i.e., all minimal modular extensions of the \(\mathbb{Z}_2^{(1)}\) anyon theory [1][5; Footnote 25]. This includes the (non-Abelian) Ising-anyon topological order [1] (a.k.a. \(p+ip\) superconducting phase [6]) as well as Abelian \(\mathbb{Z}_2\) topological order. As a subsystem code, however, the Kitaev honeycomb model does not encode any logical qubits [2].

## Encoding

## Gates

## Fault Tolerance

## Parent

- \(\mathbb{Z}_q^{(1)}\) subsystem code — The Kitaev honeycomb model can be formulated as a qubit subsystem stabilizer code [2; Sec. 3.2] based on the \(\mathbb{Z}_{q=2}^{(1)}\) Abelian anyon theory, which is non-chiral and non-modular [5; Sec. 7.3].

## Cousins

- Fermion-into-qubit code — While the Kitaev honeycomb model is bosonic, a fermion-into-qubit mapping is useful for solving and understanding the model. Logical subspace of the Kitaev honeycomb code can be formulated as a joint eigenspace of certain Majorana operators [3; Sec. 4.1]. Logical Paulis are also be constructed out of Majorana operators.
- 2D bosonization code — While the Kitaev honeycomb model is bosonic, a fermion-into-qubit mapping is useful for solving and understanding the model. This fermion-into-qubit encoding that can be converted to a 2D bosonization code by converting to a relabeled square lattice and performing single-qubit rotations [8][9; Sec. IV.B].
- Kitaev surface code — The Kitaev honeycomb model can be formulated as a qubit subsystem stabilizer code. This code can be obtained from the square-lattice surface code by gauging out the anyon \(em\) [5; Sec. 7.3]. During this process, the square lattice is effectively expanded to a hexagonal lattice [5; Fig. 12].
- Honeycomb Floquet code — The Kitaev honeycomb model Hamiltonian is a sum of checks of the honeycomb Floquet code [10].
- Matching code — Matching codes were inspired by the \(\mathbb{Z}_2\) topological order phase of the Kitaev honeycomb model [1].

## References

- [1]
- A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
- [2]
- M. Suchara, S. Bravyi, and B. Terhal, “Constructions and noise threshold of topological subsystem codes”, Journal of Physics A: Mathematical and Theoretical 44, 155301 (2011) arXiv:1012.0425 DOI
- [3]
- A. Roy and D. P. DiVincenzo, “Topological Quantum Computing”, (2017) arXiv:1701.05052
- [4]
- S. Bravyi, “Universal quantum computation with theν=5∕2fractional quantum Hall state”, Physical Review A 73, (2006) arXiv:quant-ph/0511178 DOI
- [5]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
- [6]
- F. J. Burnell and C. Nayak, “SU(2) slave fermion solution of the Kitaev honeycomb lattice model”, Physical Review B 84, (2011) arXiv:1104.5485 DOI
- [7]
- O. Raii, F. Mintert, and D. Burgarth, “Scalable quantum control and non-Abelian anyon creation in the Kitaev honeycomb model”, Physical Review A 106, (2022) arXiv:2205.10114 DOI
- [8]
- Y.-A. Chen, A. Kapustin, and Đ. Radičević, “Exact bosonization in two spatial dimensions and a new class of lattice gauge theories”, Annals of Physics 393, 234 (2018) arXiv:1711.00515 DOI
- [9]
- Y.-A. Chen and Y. Xu, “Equivalence between Fermion-to-Qubit Mappings in two Spatial Dimensions”, PRX Quantum 4, (2023) arXiv:2201.05153 DOI
- [10]
- M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021) arXiv:2107.02194 DOI

## Page edit log

- Victor V. Albert (2022-12-06) — most recent

## Cite as:

“Kitaev honeycomb code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/kitaev_honeycomb