# which are the non-abelian anyons for universal quantum computation

I am trying to get a list of non-abelian anyons that can be used for universal quantum computation by implementing gates via braiding. I found that Majorana fermions and para-fermions (not sure about all but definitely $Z_{3}$ parafermions) offer only a set of topologically protected gates but not the whole universal set of gates for quantum computation. Are Fibonacci anyons the only anyons that can lead to universal quantum computation via braiding ?

## 1 Answer

All $\mathrm{SU}(2)_k$ with $k>2, k\neq 4$ are universal. For a proof see http://arxiv.org/abs/math/0103200.

• thanks Meng Cheng; so no $Z_{n}$ parafermions are universal, just confirming ? – user56199 May 22 '15 at 5:05
• If $Z_n$ parafermions means those recent ones proposed to exist as zero modes on the edge of Abelian quantum Hall states, then yes, they are not universal. The relevant TQFT is $SO(n)_2$. But in the FQH literature, people also call $Z_k$ parafermion CFT really for $\mathrm{SU}(2)_k/U(1)$, like Read-Rezayi states. – Meng Cheng May 22 '15 at 5:11
• What is a reference for $Z_n$ parafermions and the fact that they come from metaplectic TQFTs? – Matthew Titsworth May 22 '15 at 14:13
• arxiv.org/abs/1210.5477 discussed the relation. – Meng Cheng May 22 '15 at 16:35
• @MengCheng : could you comment on this physics.stackexchange.com/questions/186336/… – user56199 May 27 '15 at 20:41