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


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 two-body interactions [1,5]. All are expected to be equivalent to each other under local Clifford transformations.



  • 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) model code — The 3F model cluster-like state encodes the temporal gate operations on the 3F subsystem code into a third spatial dimension [6]. In addition, one of possible 2D boundaries of the 3F model code is effectively a 2D 3F subsystem code.


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
T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
E. Rowell, R. Stong, and Z. Wang, “On classification of modular tensor categories”, (2009) arXiv:0712.1377
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
H. Bombin, “Topological subsystem codes”, Physical Review A 81, (2010) arXiv:0908.4246 DOI
S. Roberts and D. J. Williamson, “3-Fermion topological quantum computation”, (2020) arXiv:2011.04693
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Zoo Code ID: subsystem_three_fermion

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“Three-fermion (3F) subsystem code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021.
@incollection{eczoo_subsystem_three_fermion, title={Three-fermion (3F) subsystem code}, booktitle={The Error Correction Zoo}, year={2021}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Three-fermion (3F) subsystem code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021.