Three-fermion (3F) Walker-Wang model code[1,2] 


A 3D lattice stabilizer code whose low-energy excitations on boundaries realize the three-fermion anyon theory [35] and that can be used as a resource state for fault-tolerant MBQC [2].


3F QCA encoder [6,7], which can be simplified using bosonization [8].


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.



  • 3D bosonization code — The 3F QCA encoder [6,7] can be simplified using bosonization [8].
  • Three-fermion (3F) subsystem 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 [2].


F. J. Burnell et al., “Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order”, Physical Review B 90, (2014) arXiv:1302.7072 DOI
S. Roberts and D. J. Williamson, “3-Fermion Topological Quantum Computation”, PRX Quantum 5, (2024) arXiv:2011.04693 DOI
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
J. Haah, L. Fidkowski, and M. B. Hastings, “Nontrivial Quantum Cellular Automata in Higher Dimensions”, Communications in Mathematical Physics 398, 469 (2022) arXiv:1812.01625 DOI
J. Haah, “Topological phases of unitary dynamics: Classification in Clifford category”, (2022) arXiv:2205.09141
L. Fidkowski and M. B. Hastings, “Pumping Chirality in Three Dimensions”, (2023) arXiv:2309.15903
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Zoo Code ID: three_fermion

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