In this talk we discuss relationships between topology and quantum computation.
Since the discovery of Peter Shor's quantum algorithm for the prime factorization
of natural numbers, there has been intense interest in the discovery of new
quantum algorithms and in the construction of quantum computers. It is possible
that topology will enter in a deep way in the construction of quantum computers
based on phenomena such as the quantum Hall effect, where braiding of
quasiparticles describes unitary transformations rich enough to produce the
quantum computations.
This talk will describe the mathematics of such braiding and its relationship
with algorithms to compute topological invariants such as the Jones polynomial.
Just so, relationships with braiding go beyond the quantum Hall effect and are of
interest for constructing quantum gates and quantum algorithms. The talk will
discuss these directions and our present project in collaboration with the
research group of Prof. Glaser (on this campus) to instantiate quantum algorithms
for the Jones polynomial using NMR (Nuclear Magnetic Resonance Spectroscopy).
The talk will be self-contained both in terms of mathematics and physics.