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]. 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
Anyon initialization via quantum control [7].Gates
Clifford gates can be performed by braiding Majorana operators and Pauli measurements can be performed by measuring certain Majorana operators [3,4].CPHASE gate or a \(\pi/8\) rotation with the help of ancilla states completes a universal gate set [3,4].Fault Tolerance
One can distill ancilla states to arbitrary precision for sufficiently small noise rates and assuming perfect Clifford operations [4].Realizations
Rydberg atomic 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 [8].Superconducting qubits: driven version of the Kitaev honeycomb model [9] realized by the Pollmann group on the Sycamore and Willow devices by Google Quantum AI [10].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 [11,12]. It also allows the logical subspace of the Kitaev honeycomb model to be formulated as a joint eigenspace of certain Majorana operators [3; Sec. 4.1], which admit braiding-based gates due to their non-Abelian statistics and which can be used for topological quantum computation. 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 [13][14; Sec. IV.B].
- 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 [11,12]. It also allows the logical subspace of the Kitaev honeycomb model to be formulated as a joint eigenspace of certain Majorana operators [3; Sec. 4.1], which admit braiding-based gates due to their non-Abelian statistics and which can be used for topological quantum computation. 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 [13][14; 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 honeycomb tiling [5; Fig. 12].
- Honeycomb tiling— The Kitaev honeycomb model is defined on the honeycomb tiling.
- Topological code— The Kitaev honeycomb 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.
- Honeycomb Floquet code— The Kitaev honeycomb model Hamiltonian is a sum of checks of the honeycomb Floquet code [15].
- 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 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].
Kitaev honeycomb code
References
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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 (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