Quantum computers were first proposed by Richard Feynman and others as a way of efficiently simulating quantum mechanical processes. In 1994, Peter Shor made the subject famous by designing a quantum algorithm that breaks the widely used RSA public key encryption algorithm. Very simple quantum computers have already been constructed, but building one of useful size will be an extreme engineering challenge. One significant challenge is the elimination of decoherence errors due to environmental noise.
In 2002, Freedman, Larsen, and Wang proved that a class of proposed exotic physical theories are universal quantum computers, suggesting that manipulation of "anyons" in a two-dimensional fractional quantum Hall effect liquid could serve as the hardware layer for a quantum computer. Such an implementation would be "topological", and thus resistant to decoherence errors. The manipulation of a system of anyons can be described by a modular tensor category. Modular tensor categories and their cousins also show up in algebra, analysis and topology, and are imperfectly understood.
In this talk I'll sketch the mathematics behind quantum computing and the categorical formalism underlying topological quantum computing. I'll discuss my work in the latter area and its connection to the former.