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## "The Quantum Double Model with Boundary: Condensations and Symmetries", Salman Beigi, Peter Shor, Daniel Whalen, 2011

*Reviewed August 21, 2023*

*Citation:* Beigi, Salman, Peter W. Shor, and Daniel Whalen. "The quantum double model with boundary: condensations and symmetries." Communications in mathematical physics 306 (2011): 663-694.

*Web:* https://arxiv.org/abs/1006.5479

*Tags:* Abelian-anyons, Non-abelian-anyons

This paper gives a general theory of condensations along boundaries/domain walls in quantum double systems. It is based on the (at that point unpublished) work of Kitaev and Kong,

> Kitaev, Alexei, and Liang Kong. "Models for gapped boundaries and domain walls." Communications in Mathematical Physics 313.2 (2012): 351-373.

The main point of the paper is to demonstrate the connection between types of boundary condensations, and choices of subgroups and 2-cocycles of gauge groups.

There's a second very interesting point to this paper. When talking about groups, its important to remember that size really isn't everything. Things can be simple or complicated in different ways, and especially with small groups you often benefit from things being confidentially true.

Here is an example. Why is the group S3 so useful for topological quantum computation? Partly its what you would expect - it's the smallest non-abelian group. Partially its the fact that its actually more complicated that lots of other non-abelian groups - its also non-solvable. Partially, and this is new from this paper: its because S3 is the semidirect product of the additive and multiplicative groups of some field. Namely, F3. The additive group of F3 is Z3 and the multiplicative group is Z2, and the semidirect product of Z3 and Z2 is S3. They prove in this paper that there is a really nice general symmetry of groups of this form, which gives special structure to the methods of quantum computation built on them. The first case is F2, which recovers the toric code!

Sometimes its the properties you'd never think of which end up playing the most key roles.