Steve Flammia from the University of New Mexico has been visiting, and has been giving us a great series of tutorial talks on topological quantum computing. Steve has prepared a webpage for his tutorials (to be expanded as the tutorials go on), which you may wish to look at if you’re interested in learning the subject.
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It would be interesting if knot invariants actually ended up having an application, eg in topological quantum computing!
I havn’t had the time to properly understand topological quantum computing (being trained as a maths person and all that), however I do know about knot invariants and quantum groups etc, and the potential of topological quantum computing is very interesting!
I haven’t followed the work closely, but do know that there is a sense in which computing the Jones polynomial is complete for quantum computing. (I.e., it can be regarded as a canonical problem for quantum computing, to which all the others can be reduced). Start from quant-ph/0605181 and work backwards!
I’ll check it out when I have time – I only started understanding it in the scientific american article a few months ago – it sounded as if you could compute something, somehow, by creating a braid to encode the computation, and then calculating the Jones polynomial for the braid, which is a pretty straightforward thing to do, conceptually. (Just taking the quantum trace of a matrix).
Thanks for the reference, Michael. When I read the sci am article, I wondered about the relation between calculating a knot polynomial and using that to simulate a quantum computer, and this is discussed in this article.
Reading my previous comment – of course, doing something conceptually can be easy. Actually doing it may be difficult! (eg multiplying all the matrices representing the braidings in a braid and then taking the quantum trace of the resultant).