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## "Universal quantum computation with weakly integral anyons", Shawn Cui, Seung-Moon Hong, Zhenghan Wang, 2015

*Reviewed August 22, 2023*

*Citation:* Cui, Shawn X., Seung-Moon Hong, and Zhenghan Wang. "Universal quantum computation with weakly integral anyons." Quantum Information Processing 14 (2015): 2687-2727.

*Web:* https://arxiv.org/abs/1401.7096

*Tags:* Kitaev-quantum-double, Universal-scheme, Modular-tensor-categories

This is the paper which properly establishes universal quantum computation with the symmetric group S3.
While ostensibly this was done years earlier in a work of Mochon,

>Mochon, Carlos. "Anyon computers with smaller groups." Physical Review A 69.3 (2004): 032306.

Mochon's paper skipped a lot of details and can be seen as non-rigorous.
Interestingly enough, since the fusion space of the non-abelian anyon is three dimensional this model of quantum computation is *qutrit* based.
This can be seen as a good motivation for working in qudit-generality when writing quantum computing papers.

A key point in this paper is the detailed understanding of the braid group representations attached to group-theoretical fusion categories, as established in

> Etingof, Pavel, Eric Rowell, and Sarah Witherspoon. "Braid group representations from twisted quantum doubles of finite groups." Pacific journal of mathematics 234.1 (2008): 33-41.