Topological quantum computation - John Bryden

Michael Freedman proved that SU(2) topological quantum field theory (TQFT) is essentially equivalent to quantum computation. That is, quantum computation takes place inside a TQFT. This is demonstrated by showing that the Yang-Baxter equation of the system has a non-trivial braiding. It follows from this that the unitary transition matrices for quantum computations arise from braid group representations. The Artin braid groups are important objects of study in mathematics, understanding the representation theory of the braid groups would solve many important problems in mathematics. In particular understanding the entire representation theory of the braid groups would produce a host of new quantum invariants in dimension 3. One problem that plagues quantum computation is this lack of understanding about the unitary representation ring of the braid groups. Part of my research is devoted to understanding the representation theory of the braid groups through the application of stable homotopy theory in topology. This talk will outline Freedman's idea for constructing unitary matrices necessary for quantum computation using 0+1 dimensional topological quantum field theory.