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Quasiparticles theoretically can form braids on a 2D surface. The braids apparently are quite stable for reasonably long periods, allowing a superposition state more time before it decoheres through environmental interaction. This would be a big step in quantum computing. Has this actually been achieved or is it theory only at this point?

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Assuming anyons exist in fractional quantum Hall states (see Deepak's answer and its comments for very good theoretical and experimental reasons that they do), anybody who has made a fractional quantum Hall system has indeed created braided anyons. We are still quite far from actually being able to dynamically braid the anyons, or even definitively experimentally observe the anyons, both of which, in some form, are necessary to do quantum computation (topological quantum computation by measurement gets around actually having to braid the anyons, but in some sense it braids them virtually). –  Peter Shor Feb 24 '11 at 11:29
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So far non-abelian quasiparticles have yet to be observed in quantum hall states. However, as far as theory is concerned, in a recent paper a rigorous proof is provided that such quasiparticles do exist in quantum hall states. There were many candidate states for this object, starting with the Moore-Read state, however, theoretically there did not yet exist a complete proof of their existence.

This is a really big achievement. Frank Wilzcek describes it as "a landmark proof".

There have been various incompletely verified "sightings" of the $5/2$ state which is the non-abelian state. For results upto 2008 at least, see the RMP paper on topological quantum computation. At the end of this paper various references are given which claim to have observed this state. I don't know what the latest is, but I would have heard of an definitive experimental observation, or so I like to think :)

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Not super recent, but perhaps you're aware of Bob Willett's experiments (0911.0345 and 0807.0221)? Not a smoking gun by any means, but cute and very encouraging. –  wsc Feb 24 '11 at 0:45
    
Thanks for pointing those out. –  user346 Feb 24 '11 at 2:46
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