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Shinn497
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Hello all,
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-DSecondary 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?
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-DSecondary 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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