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The first thing to do is to think topologically: make sure you understand why a coffee cup is the same thing topologically as a donut. Now, imagine we swap two identical particles, and do it again, so that we are back where we started. Apply this topological thinking to the paths taken by the particles: it is the same as doing nothing. Here I show a ...

6

It depends what you mean by the 'existence' of anyons. One way is to engineer a Hamiltonian which leads to quasiparticles (or other defects) that have anyonic statistics. This will require the Hamiltonian to be implemented, the system to be cooled to sufficiently near the ground state, the anyons to be manipulated, etc. So there's a lot to be done, and I ...

5

The Toric code is an error correcting code. The distance of the code (I.e. the number of local operations required to convert one logical state into an orthogonal one) is equal to $N$, where the Toric code is defined on an $N\times N$ grid. One of the places that the performance of the Toric code really wins out is that although it is only distance $N$, the ...

5

Are there other instances of topological QC that do not use anyons? No, that's basically by definition. That said, there are different ways that one could use topological systems in order to achieve quantum computation. In the version you're talking about, you use these anyon pairs to define qubits, and braid them around each other to create quantum gates. ...

3

The spin-statistics theorem requires a particle's wave function to acquire the same phase when it is rotated by an angle of $2 \pi$ about itself and when exchanged with an identical (indistinguishable) particle. For example, a fermion acquires a phase of $-1$ both in exchange and self-rotation. Thus, when we think of an anyon as a flux-charge entity, we ...

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