Three-fermion (3F) model code[1] 


A 3D topological code whose low-energy excitations realize the three-fermion anyon theory [24] and that can be used as a resource state for fault-tolerant MBQC [1].


Clifford gates can be performed by braiding and fusing symmetry defects in the MBQC model.

Fault Tolerance

Fault-tolerant MBQC protocol by encoding in, braiding, and fusing symmetry defects.



  • Three-fermion (3F) subsystem code — The 3F model cluster-like state encodes the temporal gate operations on the 3F subsystem code into a third spatial dimension [1]. In addition, one of possible 2D boundaries of the 3F model code is effectively a 2D 3F subsystem code.


S. Roberts and D. J. Williamson, “3-Fermion topological quantum computation”, (2020) arXiv:2011.04693
E. Rowell, R. Stong, and Z. Wang, “On classification of modular tensor categories”, (2009) arXiv:0712.1377
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
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
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Zoo Code ID: three_fermion

Cite as:
“Three-fermion (3F) model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.
  title={Three-fermion (3F) model code},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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Cite as:

“Three-fermion (3F) model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023.