Three-fermion (3F) subsystem code[1,2] 

Description

2D subsystem stabilizer code whose low-energy excitations realize the three-fermion anyon theory [1,3,4]. One version uses two qubits at each site [2], while other manifestations utilize a single qubit per site and only weight-two (two-body) interactions [1,5]. All are expected to be equivalent to each other via a local constant-depth Clifford circuit.

Parents

Cousins

  • Kitaev surface code — One version of the 3F subsystem code can be obtained from two copies of the square-lattice surface code by gauging out the anyons \(e_1m_1e_2\) and \(e_2m_2\) [2; Sec. 7.4].
  • Three-fermion (3F) Walker-Wang model code — The (three-dimensional) 3F Walker-Wang model cluster-like state encodes the temporal gate operations on the (two-dimensional) 3F subsystem code into a third spatial dimension [6].
  • 2D color code — The 2D color code is equivalent to two decoupled copies of the 3F code in the sense that the same anyon theory describes the low-energy excitations of both codes [8][7; Appx. B].

References

[1]
H. Bombin, M. Kargarian, and M. A. Martin-Delgado, “Interacting anyonic fermions in a two-body color code model”, Physical Review B 80, (2009) arXiv:0811.0911 DOI
[2]
T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
[3]
E. Rowell, R. Stong, and Z. Wang, “On classification of modular tensor categories”, (2009) arXiv:0712.1377
[4]
H. Bombin, G. Duclos-Cianci, and D. Poulin, “Universal topological phase of two-dimensional stabilizer codes”, New Journal of Physics 14, 073048 (2012) arXiv:1103.4606 DOI
[5]
H. Bombin, “Topological subsystem codes”, Physical Review A 81, (2010) arXiv:0908.4246 DOI
[6]
S. Roberts and D. J. Williamson, “3-Fermion Topological Quantum Computation”, PRX Quantum 5, (2024) arXiv:2011.04693 DOI
[7]
M. S. Kesselring et al., “The boundaries and twist defects of the color code and their applications to topological quantum computation”, Quantum 2, 101 (2018) arXiv:1806.02820 DOI
[8]
Zhenghan Wang. private communication, 2017.
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Zoo Code ID: subsystem_three_fermion

Cite as:
“Three-fermion (3F) subsystem code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/subsystem_three_fermion
BibTeX:
@incollection{eczoo_subsystem_three_fermion, title={Three-fermion (3F) subsystem code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/subsystem_three_fermion} }
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“Three-fermion (3F) subsystem code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/subsystem_three_fermion

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