Description
Subsystem qubit stabilizer code underlying the Kitaev honeycomb model [1,2]. Its gauge generators are the two-qubit \(XX\), \(YY\), and \(ZZ\) link operators on the three edge types of the honeycomb lattice [2; Sec. 3.2]. Its stabilizer group is generated by loop operators, and syndrome extraction can be reduced to ordered measurements of the two-qubit link operators [2; Sec. 3.2]. This is the \(q=2\) instance of the \(\mathbb{Z}_q^{(1)}\) subsystem code and does not encode any logical qubits [2][3; Sec. 7.3].
The original Kitaev honeycomb spin model is exactly solvable by mapping spins to Majorana fermions in a static \(\mathbb{Z}_2\) gauge field, yielding three gapped \(A\) phases and one gapless \(B\) phase [1]. Its ground state lies in the vortex-free sector, and the gapped \(A\) phases realize Abelian \(\mathbb{Z}_2\) topological order [1].
Encoding
The geometric entanglement measure of a ground state of the Kitaev honeycomb model and any state with anomalous one-form symmetry scales as order \(\Omega(n)\) [4].Realizations
Neutral atom arrays: realized on a 72 qubit device with 32 ancillas by the Lukin group, where a fermion-into-qubit mapping was used to recast this model in terms of simulated fermionic degrees of freedom and simulate other fermionic Hamiltonians [5].Superconducting qubits: driven version of the Kitaev honeycomb model [6] realized by the Pollmann group on the Sycamore and Willow devices by Google Quantum AI [7].Cousins
- Tetron code— Embedding each physical qubit into two fermions via the tetron code is useful for exactly solving the Kitaev honeycomb model Hamiltonian [1] and other qubit Hamiltonians on certain graphs [8,9].
- 2D bosonization code— Embedding each physical qubit into two fermions via the tetron code is useful for exactly solving the Kitaev honeycomb model Hamiltonian [1] and other qubit Hamiltonians on certain graphs [8,9]. When done in reverse, this embedding can be thought of as a 2D bosonization fermion-into-qubit encoding by converting to a relabeled square lattice and performing single-qubit rotations [10][11; Sec. IV.B].
- Kitaev surface code— The Kitaev honeycomb 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 honeycomb tiling [3; Fig. 12].
- Honeycomb tiling— The Kitaev honeycomb code is defined on the honeycomb tiling.
- Honeycomb Floquet code— The Kitaev honeycomb model Hamiltonian is a sum of checks of the honeycomb Floquet code [12].
- Non-Abelian Kitaev honeycomb code— The gauge-group generators of the Kitaev honeycomb code are terms of the Kitaev honeycomb model Hamiltonian. Adding a magnetic field to this Hamiltonian for particular parameter values yields the non-Abelian Ising-anyon phase, whose anyons encode the logical information of the non-Abelian Kitaev honeycomb code [1].
- Matching code— Matching codes were inspired by the \(\mathbb{Z}_2\) topological order phase of the Kitaev honeycomb model [1].
- 3D Kitaev honeycomb code— The 3D Kitaev honeycomb model is a 3D generalization of the Kitaev honeycomb model.
Primary Hierarchy
Parents
\(\mathbb{Z}_q^{(1)}\) subsystem codeAbelian topological Topological Hamiltonian-based QECC Lattice subsystem Subsystem QECC Quantum
The Kitaev honeycomb code is the \(q=2\) instance of the \(\mathbb{Z}_q^{(1)}\) subsystem code [3; Sec. 7.3].
Kitaev honeycomb code
References
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- T. D. Ellison, Y.-A. Chen, A. Dua, W. Shirley, N. Tantivasadakarn, and D. J. Williamson, “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
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- S. J. Evered et al., “Probing the Kitaev honeycomb model on a neutral-atom quantum computer”, Nature 645, 341 (2025) arXiv:2501.18554 DOI
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- H. C. Po, L. Fidkowski, A. Vishwanath, and A. C. Potter, “Radical chiral Floquet phases in a periodically driven Kitaev model and beyond”, Physical Review B 96, (2017) arXiv:1701.01440 DOI
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- M. Will, T. A. Cochran, E. Rosenberg, B. Jobst, N. M. Eassa, P. Roushan, M. Knap, A. Gammon-Smith, and F. Pollmann, “Probing non-equilibrium topological order on a quantum processor”, Nature 645, 348 (2025) arXiv:2501.18461 DOI
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- A. Chapman and S. T. Flammia, “Characterization of solvable spin models via graph invariants”, Quantum 4, 278 (2020) arXiv:2003.05465 DOI
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- S. J. Elman, A. Chapman, and S. T. Flammia, “Free Fermions Behind the Disguise”, Communications in Mathematical Physics 388, 969 (2021) arXiv:2012.07857 DOI
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- 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
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- Y.-A. Chen and Y. Xu, “Equivalence between Fermion-to-Qubit Mappings in two Spatial Dimensions”, PRX Quantum 4, (2023) arXiv:2201.05153 DOI
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- M. B. Hastings and J. Haah, “Dynamically Generated Logical Qubits”, Quantum 5, 564 (2021) arXiv:2107.02194 DOI
Page edit log
- Victor V. Albert (2026-04-08) — most recent
Cite as:
“Kitaev honeycomb code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/kitaev_honeycomb