Quantum computer: Feynman's idea is, broadly, to use the quantum nature of physical systems to build a massively parallel computer. There are several major technical obstructions to puting this into practice. One is that this device breaks the discreteness of binary computation (a barrier to errors in the classical case), so problems have to be recoded to make computation computational-fault-tolerant.
One kind of discreteness that can be set up (in principle) to survive the quantum setting, is topological discreteness. (Very roughly speaking, while the precise trajectories of a collection of particles in a plane from one time to another are obscured from us by system noise, the braiding of suitably prepared particle time-lines is a relatively robust datum.) This leads us to study certain realisations/representations of braid groups, such as those provided by the Temperley-Lieb algebra.
The TL algebra gives a microcosm of our methodology. It and its family can be studied from a number of different perspectives. From the representation theory perspective, the fundamental invariants of TL (and several other such algebras) over the complex field have been worked out by members of the Leeds AGIS group and their collaborators. (Our original motivation was applications in statistical mechanics. But the algebraic framework seems well suited, in principle, to mathematical studies of QI as well. In actuality, of course, the contribution of our old representation theory results to QI is epsilonic at best... but it does seem like a good excuse to get involved.)
Leeds: Jiannis Pachos, Paul Martin, Joao Faria Martins, Frank Nijhoff, Alex Bullivant, Celeste Damiani, Fiona Torzewska, Kon Meichanetzidis (now Oxford), Tim Spiller (now York), Zoltan Kadar, Giandomenico Palumbo (now Brussels), ...
September 2012 - April 2018         Go here to read more.