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A topological quantum computer is a theoretical quantum computer that employs two-dimensional quasiparticles called anyons. -Wikipedia
Are there other instances of topological quantum computing models that do not use anyons?
Are there alternative forms of anyons besides Fibonacci anyons?
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. Another option is more related to quantum memories: topological systems have a degenerate ground state that can be used to encode a qubit. But that qubit should be very robust against noise because it takes many many single-qubit errors (spanning the bulk of the system) to be misinterpreted as a logical operation. If you have many of these, you could use them as qubits in a quantum computer, but the challenge is actually getting them to do gates.
Are there alternative forms of anyons besides Fibonacci anyons?
There's a huge variety, it's just that Fibonacci anyons are a universal non-Abelian type of anyon that's comparatively easy to explain, but one does not have to be restricted to them if you can prove the right properties in some other system. The other type that are commonly discussed are Ising anyons.
Basically, every quantum system with localisable excitations has anyons, but their properties vary wildly. For example, if you look at Kitaev's Toric code, there are two types of anyon present. The particle-anti-particle pairs are basically joined together either by a string of $X$ operators, or a string of $Z$ operators. You move them around by applying $X$s or $Z$s as appropriate. When you braid them past each other, there's a single site where an $X$ and a $Z$ coincide. Since these two operators anti-commute, that corresponds to acquiring a -1 phase. Thus, the braiding effectively implements a logical $Z$. But that's all it implements; it can't change the type of excitation that the system is in (hence, we say it has Abelian anyons). That makes the computations you can do with them extremely limited. Even if you can find non-Abelian anyons, they may be of a limited form which is not universal for quantum computation (you might get an equivalent of the Clifford gates, for example).
Are there other instances of topological QC that do not use anyons?
The use of anyons is part of the definition of topological QC.
Are there alternative forms of anyons besides Fibonacci anyons?
There are Fibonacci anyons and Ising anyons. An excellent reference is Non-Abelian anyons: when Ising meets Fibonacci.
