Non-Abelian Kitaev honeycomb code[1]
Description
Code whose logical subspace in the gapped non-Abelian phase of the Kitaev honeycomb model with a magnetic field is labeled by different fusion outcomes of Ising anyons [1].
The original Kitaev honeycomb spin model is exactly solvable by mapping spins to Majorana fermions in a static \(\mathbb{Z}_2\) gauge field (equivalently, embedding each physical qubit into two fermions via the tetron code [2; Sec. 4.1]), yielding three gapped \(A\) phases and one gapless \(B\) phase [1]. A magnetic field opens a gap in phase \(B\) of the underlying Kitaev honeycomb code and yields the non-Abelian Ising-anyon phase [1] (a.k.a. \(p+ip\) superconducting phase [3]). In the honeycomb model with magnetic field, the spectral Chern number is \(\nu=\pm 1\) depending on the field direction; more generally, gapped free-fermion phases with \(\mathbb{Z}_2\) vortices are classified by a spectral Chern number \(\nu\), and their anyonic properties depend on \(\nu \bmod 16\) [1].
Ising anyons also exist in other phases, such as the fractional quantum Hall phase [4].
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 [2,4].CPHASE gate or a \(\pi/8\) rotation with the help of ancilla states completes a universal gate set [2,4].Fault Tolerance
One can distill ancilla states to arbitrary precision for sufficiently small noise rates and assuming perfect Clifford operations [4].Cousins
- 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].
- Tetron code— Embedding each physical qubit into two fermions via the tetron code allows the logical subspace of the Kitaev honeycomb model to be formulated as a joint eigenspace of certain Majorana operators [2; 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 [6][7; Sec. IV.B].
- 2D bosonization code— Embedding each physical qubit into two fermions via the tetron code allows the logical subspace of the Kitaev honeycomb model to be formulated as a joint eigenspace of certain Majorana operators [2; 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 [6][7; Sec. IV.B].
- Honeycomb tiling— The Kitaev honeycomb model is defined on the honeycomb tiling.
Primary Hierarchy
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]
- 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
- [4]
- S. Bravyi, “Universal quantum computation with theν=5∕2fractional quantum Hall state”, Physical Review A 73, (2006) arXiv:quant-ph/0511178 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]
- 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
- [7]
- Y.-A. Chen and Y. Xu, “Equivalence between Fermion-to-Qubit Mappings in two Spatial Dimensions”, PRX Quantum 4, (2023) arXiv:2201.05153 DOI
Page edit log
- Victor V. Albert (2022-12-06) — most recent
Cite as:
“Non-Abelian Kitaev honeycomb code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/nonabelian_kitaev_honeycomb