Anyons
As we discussed before, anyons are special types of particles. They exist only in two dimensions and have unique properties. When anyons are swapped or "braided" around each other, it changes their quantum state.
Non-Abelian Anyons
There are two types of anyons, Abelian and non-Abelian. Without getting into the details, the important thing to know is that non-Abelian anyons have an extra special property: when you swap them around, the order in which you do it matters. This is not the case with Abelian anyons or with most particles, we're familiar with.
Encoding Information
In the context of quantum computing, "encoding information" means to represent information in the form of quantum states. For a regular qubit, this would be the 0 and 1 states and their superpositions. For anyons, this information is encoded in their braiding patterns.
The fact that in a topological quantum computer, the quantum information is not stored in individual anyons, but rather in the overall pattern of their braiding. This means that the information is in some sense "distributed" across the system, rather than being located at a specific point. This distribution of information is part of what makes topological quantum computers resistant to errors.
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