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I'm working through the following paper on topological quantum computing.

http://www.qip2010.ethz.ch/tutorialprogram/JiannisPachosLecture

In particular I'm trying to derive and solve the pentagon equation in order to evaluate the F matrices for ising anyons. One thing I'm trying to determine is how to determine, in general, which fusion paths can become a 2-d hilbert space. I understand that the fusion paths in figure 1.4a are analogues of the |0> |1>, and |+> |-> hilbert spaces but I don't get if there is a way to find any others...in general.

I think what would really help is equation 1.6, which determines the dimensional of of a set of anyons based on the fusion rules. BUT I'm not entirely sure how to read that equation. In particular I can't get how it is used to properly get the dimension of 3 σ anyons to be 2. I was able to prove it by drawing it out and doing the fusions but I couldn't use that one equation.

This is important because the pentagon identity is for 4 anyons. This means there are a 243 possible fusion paths. In the intermediate steps of the equation there are more. If there was a better way of determining which fusion paths are within the fusion rules that would be great.

Thanks for your help!

EDIT: I meant to say hilbert spaces in general they do not have to be 2-D

Secondary question:

Why does braiding effect the fusion? This is the central idea to what makes quantum computation with anyons possible but I'm not getting it. I'll accept that anyons gain a phase when braided but I'm not sure how this phase effects the fusion outcome in general. In other words, can you explain figure 1.3?

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# Determining how to find the 2-d hilbert space from fusing ising anyons

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