Kitaev honeycomb code

Kitaev honeycomb code[1,2]

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

Anyon initialization via quantum control [5].

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.
\(\mathbb{Z}_q^{(1)}\) 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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/subsystem/topological/kitaev_honeycomb.yml.