Lecturer: Pawel Wocjan, School of Computer Science, University of Central Florida

I will explain relationships between the Jones polynomial and quantum computation. First, I will present polynomial-time quantum algorithms which give additive approximations of the Jones polynomial of any link obtained from a certain general family of closures of braids, evaluated at any primitive root of unity. Our algorithms are based on a local qubit implementation of the unitary Jones-Wenzl representations of the braid group which makes the underlying representation theory apparent. Next, I will outline our proof that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof uses a natural encoding of two-qubit unitaries into the rectangular representation of the eight-strand braid group. |