Anyons in Quantum Computation

Quantum computation is unpleasantly susceptible to environmental disturbances. Its advantages over classical computation depend on maintaining superpositions of state vectors, with high precision in the coefficients of those vectors. Small disturbances can easily modify those coefficients or, indeed, destroy superpositions altogether. Significant effort must therefore be devoted to error correction, and this makes algorithms slower and harder to design.

It has been suggested [6] that qubits could be more robust, i.e., less susceptible to disturbances, if they were implemented using certain sorts of anyons. For example, if qubits were encoded in the way two anyons wind around each other, then this winding, being a topological property of the system, would be robust. A small disturbance in the actual motion of the anyons would leave the winding number intact. This hope of reducing the error correction needs of quantum computing has motivated much of the current interest in anyons.

In this approach to quantum computation, braiding of anyons serves not only to store information but also to process it. In general, as mentioned above, quantum computation proceeds by initializing a quantum state, then applying a unitary transformation to it, and finally measuring some observable in the resulting transformed state. The unitary transformation used here must be designed so that a feasible measurement produces a useful result. In addition, there must be a way to implement the unitary transformation as the composition of a sequence of simpler unitary transformations, usually called gates in this context. In the anyon approach to quantum computation, the most basic unitary gates arise from the braiding of anyons around each other, and a crucial question is whether these gates are universal in the sense that arbitrary gates can be approximated by composing the basic ones.

It is worth noting explicitly that, in this picture, a qubit is not encoded in the state of a single anyon but rather in a whole system of several anyons. This feature will be quite prominent in the category picture described in the rest of this paper.

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