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## "3-Fermion topological quantum computation", Sam Roberts, Dominic Williamson, 2020

*Reviewed August 21, 2023*

*Citation:* Roberts, Sam, and Dominic J. Williamson. "3-fermion topological quantum computation." arXiv preprint arXiv:2011.04693 (2020).

*Web:* https://indico.physik.uni-muenchen.de/event/84/attachments/244/440/S1A.Roberts.abstract.pdf

*Tags:* Higher-dimensional, Abelian-anyons, Universal-scheme

This paper shows that the (abelian) "3-fermion" MTC obtains the full Clifford group by braiding alone, if you store information in the correct way.
Their model works in (3+1) dimensions, fitting into the general "Walker-Wang" paradigm:

> Walker, Kevin, and Zhenghan Wang. "(3+ 1)-TQFTs and topological insulators." Frontiers of Physics 7 (2012): 150-159.

This paper contains a lot of interesting ideas, and is a good reference for the 3-fermion model,
which become a staple of the field. The original reference for the 3-fermion model goes back to 2009:

> Bombin, Hector, M. Kargarian, and M. A. Martin-Delgado. "Interacting anyonic fermions in a two-body color code model." Physical Review B 80.7 (2009): 075111.

The original 3 fermion model lives in 2+1 dimensions. It it is a theory with 4 simple objects. All three non-trivial simple objects
are fermions, hence the name. This theory also goes by the name of the (D4,1) model, as included in the table in

> Rowell, Eric, Richard Stong, and Zhenghan Wang. "On classification of modular tensor categories." Communications in Mathematical Physics 292.2 (2009): 343-389.

A nice Hamiltonian realization of the (2+1)-dimensional dimensional 3 fermion model is given in

> Bombín, Héctor. "Topological subsystem codes." Physical review A 81.3 (2010): 032301.

The fact that this paper deals also with (3+1)-dimensional is not very important,
and much of the paper also deals with (2+1)-dimensional 3-fermionic phenomena.