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## "Discrete non-Abelian gauge theories in Josephson-junction arrays and quantum computation", Benoit Doucot, Lev Ioffe, Julien Vidal, 2004

*Reviewed October 6, 2023*

*Citation:* Douçot, Benoit, Lev B. Ioffe, and Julien Vidal. "Discrete non-Abelian gauge theories in Josephson-junction arrays and quantum computation." Physical Review B 69.21 (2004): 214501.

*Web:* https://arxiv.org/abs/cond-mat/0302104

*Tags:* Pedagogical, Kitaev-quantum-double, Hardware, xy-anyons

This paper is a very early take on topological quantum computation
in the language of gauge theory, which is more grounded than its contemporaries. It considers explicit lattice models
with Josephson-junction arrays which are equivalent to discrete non-Abelian gauge theories.
These situations host non-abelian anyons which are well described. This paper includes
a lovely review of relevant material, and lots of pretty pictures.

Another nice feature of this paper is the surrounding discussion. For instance,
in the conclusion the authors raise a very interesting question:
"It is not clear whether it is absolutely unavoidable to emulate gauge theory in order
to get the topological protection and non-Abelian statistics. What is the most general class of theories that has
these properties?". Perhaps one answer to this question is the existence
of the Levin-Wen model, based on arbitrary spherical fusion categories.
Group-theoretical spherical fusion categories (i.e. Vec_G for some group G)
correspond to models emulating gauge theories, but arbitrarily fusion categories
don't have such naturally associated gauge groups. Some MTCs come from
gauge theories in more subtle ways, like quantum group MTCs. There are
MTCs which don't come from quantum groups however, and instead come
from more exotic vertex operator algebras.

Another question raised is whether or not Kosterlitz-Thouless type
transitions appear in these non-abelian lattice settings. I have no idea.
An interesting reference is a paper in which the Kosterlitz-Thouless transition
is demonstrated very nicely theoretically in a lattice based model:

> Park, Kyungwha, and David A. Huse. "Superconducting phase with fractional vortices in the frustrated kagom é wire network at f= 1/2." Physical Review B 64.13 (2001): 134522.

Another reason to cite this paper: It is an early example of experimentally-minded
people understanding that the dihedral group D_4 is the simplest non-abelian group,
*not* S_3. Their non-abelian models focus entirely on the dihedral groups
D_4 and D_5.