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Anyon models do not lead in general to universal topological quantum computation (= existence of a universal set of quantum gates) when only the braiding is used for implementing gates. The Fibonacci anyon model has been shown to be universal for quantum computation (see e.g. http://www.thp.uni-koeln.de/trebst/pubs/FibonacciAnyonModels.pdf for an overview), as are all $\mathrm{SU}(2)_k$-models with $k>2$, $k\neq 4$ (https://arxiv.org/abs/math/0103200), while the Ising model is not (in https://arxiv.org/abs/1208.4834 it is shown that a bilayer Ising model can be made universal when introducing permutation defects).

Question: Are there any other known anyon models besides the Ising model which are non-universal when only the braiding is used? I am particularly interested in models of non-abelian anyons. (Please provide literature sources when answering this question!)

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    $\begingroup$ In general, abelian anyon models are not universal. $\endgroup$ – AccidentalFourierTransform May 10 '18 at 13:39
  • $\begingroup$ @AccidentalFourierTransform What about non-abelian anyons? And could you please state the research papers involved? $\endgroup$ – kolaka May 10 '18 at 17:54
  • $\begingroup$ see e.g. arxiv.org/abs/0904.4373 $\endgroup$ – AccidentalFourierTransform May 10 '18 at 19:19
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For the case of non-abelian anyons, the property F conjecture (http://www.math.tamu.edu/~rowell/naidurowell3.pdf) would imply that any weakly integral anyon theory (i.e. one where all quantum dimensions are square roots of integers) is not universal.

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  • $\begingroup$ Thanks, that is particularly helpful as this is a property on the level of modular tensor categories. Please elaborate if you have some more information about this to share! $\endgroup$ – kolaka Jun 15 '18 at 10:05

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