## Description

Code whose logical subspace is labeled by different fusion outcomes of Ising anyons present in the nonabelian 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 nonabelian statistics and which can be used for topological quantum computation.

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 [3; 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][3; Footnote 25].

The Kitaev honeycomb code utilizes the nonabelian Ising-anyon topological order of the Kitaev honeycomb model [1] (a.k.a. \(p+ip\) superconducting phase [4]) as well as abelian \(\mathbb{Z}_2\) topological order. More generally, the Hamiltonian realizes all anyon theories of the 16-fold way, i.e., all minimal modular extensions of the \(\mathbb{Z}_2^{(1)}\) abelian non-chiral non-modular anyon theory [1][3; Footnote 25].

## Encoding

## Gates

## Parents

- Majorana stabilizer code — While the Kitaev honeycomb model is bosonic, a fermionic 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 [2; Sec. 4.1]. Logical Paulis are also be constructed out of Majorana operators.
- \(\mathbb{Z}_q^{(1)}\) subsystem code — The Kitaev honeycomb model can be formulated as a qubit subsystem stabilizer code based on the \(\mathbb{Z}_{q=2}^{(1)}\) abelian anyon theory, which is non-chiral and non-modular [3; Sec. 7.3].

## Cousins

- 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 \(em\) [3; Sec. 7.3]. During this process, the square lattice is effectively expanded to a hexagonal lattice [3; Fig. 12].
- Honeycomb Floquet code — The Kitaev honeycomb model has a Hamiltonian which is the sum of checks of the honeycomb Floquet code [6].
- 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]
- A. Roy and D. P. DiVincenzo, “Topological Quantum Computing”, (2017) arXiv:1701.05052
- [3]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, (2022) arXiv:2211.03798
- [4]
- 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
- [5]
- 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
- [6]
- 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