Instead of picking up a phase of -1 (like Fermions) or 1 (like Bosons), swapping Anyon's around has to be represented by a unitary operation which results in a

Instead of picking up a phase of -1 (like Fermions) or 1 (like Bosons), swapping Anyon's around has to be represented by a unitary operation which results in a superposition state (I don't fully understand this myself). The swapping unitary is universal for quantum computing. Hence proposals to use the "braiding" of anyons (re: shunting them around on guides) to build a quantum computer. They can in theory move around as much as they want and so long as they don't swap no operation is effectively performed. Hence they are thought to be a robust choice of qubit. Actually building them has proven very hard in practice. AFAIK most researchers think it is a dead end.