## Description

Code whose logical subspace is labeled by different fusion outcomes of Ising anyons present in the non-Abelian 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.

## Encoding

Anyon initialization via quantum control [3].

## Gates

Clifford gates can be performed by braiding Majorana operators and Pauli measurements can be performed by measuring certain Majorana operators [2].CPHASE gate or a \(\pi/8\) rotation with the help of ancilla states completes a universal gate set [2].

## 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.
- Topological code — Kitaev honeycomb codes utilizes the non-Abelian Ising-anyon topological order phase of the Kitaev honeycomb model [1]. This phase is also known as a \(p+ip\) superconducting phase [4]. The Kitaev honeycomb model also admits an abelian \(\mathbb{Z}_2\) topological order.

## Cousins

- Honeycomb Floquet code — The Kitaev honeycomb model has a Hamiltonian which is the sum of checks of the honeycomb Floquet code [5].
- 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]
- 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
- [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]
- 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