I have been trying to get a basic idea of what anyons are for the past couple of days. However, the online articles (including Wikipedia) seem unusually vague and impenetrable as far as explaining topological quantum computing and anyons goes.
The Wiki page on Topological quantum computer says:
A topological quantum computer is a theoretical quantum computer that employs two-dimensional quasiparticles called anyons, whose world lines pass around one another to form braids in a three-dimensional spacetime (i.e., one temporal plus two spatial dimensions). These braids form the logic gates that make up the computer. The advantage of a quantum computer based on quantum braids over using trapped quantum particles is that the former is much more stable. Small, cumulative perturbations can cause quantum states to decohere and introduce errors in the computation, but such small perturbations do not change the braids' topological properties.
This sounded interesting. So, on seeing this definition I tried to look up what anyons are:
In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than fermions and bosons. In general, the operation of exchanging two identical particles may cause a global phase shift but cannot affect observables.
Okay, I do have some idea about what quasiparticles are. For example, as an electron travels through a semiconductor, its motion is disturbed in a complex way by its interactions with all of the other electrons and nuclei; however, it approximately behaves like an electron with a different mass (effective mass) travelling unperturbed through free space. This "electron" with a different mass is called an "electron quasiparticle". So I tend to assume that a quasiparticle, in general, is an approximation for the complex particle or wave phenomenon that may occur in matter, which would be difficult to mathematically deal with otherwise.
However, what do they mean by "much less restricted than fermions and bosons"? Do "anyons" follow a different kind of statistical distribution than what bosons or fermions follow?
In the next line, they say that exchanging two identical particles may cause a global phase shift but cannot affect the observables. What is meant by global phase shift in this context? Moreover, which observables are they actually talking about here?
How are these quasiparticles i.e. anyons actually relevant to quantum computing? I keep hearing vague things likes "The world-lines of anyons form braids/knots in 3-dimensions (2 spatial and 1 temporal). These knots help form stable forms of matter, which aren't easily susceptible to decoherence". I think that this Ted-Ed video gives some idea, but it seems to deal with restricting electrons (rather than "anyons") to move on a certain closed path inside a material.
I would be glad if someone could help me to connect the dots and understand the meaning and significance of "anyons" at an intuitive level. I think a layman-level explanation would be more helpful for me, initially, rather than a full-blown mathematical explanation. However, I do know basic undergraduate level quantum mechanics, so you may use that in your explanation.