Algebraic, geometric and physical underpinnings of topological quantum computation

Summary

The motivating aim is to build a quantum computer. Feynman's idea is, broadly, to use the quantum nature of physical systems to build a massively parallel computer. There are several major technical obstructions to puting this into practice. One is that this device breaks the discreteness of binary computation (a barrier to errors in the classical case), so problems have to be recoded to make computation computational-fault-tolerant.

One kind of discreteness that can be set up (in principle) to survive the quantum setting, is topological discreteness. (Very roughly speaking, while the precise trajectories of a collection of particles in a plane from one time to another are obscured from us by system noise, the braiding of suitably prepared particle time-lines is a relatively robust datum.) This leads us to study certain realisations/representations of braid groups, such as those provided by the Temperley-Lieb algebra, and several beautiful generalisations of this type of problem.

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