# Topological quantum computation: abelian vs. non-abelian anyons

We need non-abelian fractional hall states because of the ground state degeneracy http://rmp.aps.org/abstract/RMP/v80/i3/p1083_1 (arXiv version for free). But we can also have degeneracy even in case of abelian fractional quantum hall states on topologcally non-trivial surfaces http://prb.aps.org/abstract/PRB/v41/i13/p9377_1. Could anyone tell me the reason (conceptual) for not using abelian state for quantum computation?

• We won't have universal computation. – Ali Jul 19 '13 at 9:25
• Welcome to SE.physics. When it's possible, please prefer to give the arXiv, free version of the paper you cite. I edited your question. It's about abelian vs. non-abelian anyons. – FraSchelle Jul 19 '13 at 9:26

You need some criteria to perform computation at the quantum level. One of them is the ability to perform any manipulation of your qubit. Abelian anyons does not provide full possibility to manipulate the qubit state (in particular you can not rotate the phase the way you want, only by $\pi/4$ for instance for Majorana modes in superconducting wires, and the braiding is then called "abelian braiding").