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It has been known for some time that quantum computation is equivalent to approximating the Jones polynomial at certain roots of unity. This result, due to Friedman, Kitaev, Larsen, Wang, was presented in the language of topological quantum field theories and was not widely understood by the quantum computing community.

In this talk we'll explain some aspects of the connection between the Jones polynomial and quantum computation without using the language of topological quantum field theory. We'll present some recent work (with D. Aharonov and V. Jones) that gives an explicit quantum algorithm for approximating the Jones polynomial. We'll also explore some of the directions for future research that this connection opens up.

No detailed knowledge of any of the words appearing in the title will be assumed.