**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.