Abstract for Lecture I:

I will start with an overview of axiomatic quantum mechanics, and introduce the quantum circuit model of quantum computing. Then I will explain the difficulty in building a large scale quantum computer. Following that I will outline how to use the Jones representations of the braid groups to build topological a model of quantum computers. It follows that the computational complexity of quantum computing is related to that of the computation of the Jones polynomial of links. Finally I will mention a physical candidate for the realization of topological quantum computing: fractional quantum Hall liquids.

Why topology is helpful? The major obstacle to building a quantum computer is error correction. While there is a beautiful theoretical solution using error correction codes, a physical realization following this path is daunting, if not impossible. On the other hand, many global topological properties are, by definition, invariant under deformation, therefore information encoded in topological properties is robust against most errors since most errors are local. Although it is not obvious, one example of a topological property of this type is the Fermi statistics of particles: when two electrons are exchanged, their wavefunction acquires a phase factor -1, which is independent of the details of the exchanging path. In 3 and higher dimensions, all particles are either fermions or bosons. But in dimension 2, there exist quasi-particles with statistics between those of fermions and bosons: the so-called anyons. Anyons are quasi-particles having the property that when two of them are exchanged, their wavefunction acquires a phase factor other than 1 or –1. More interestingly, the wavefunction of several anyons can be changed by a unitary matrix when two of them are exchanged. In fact, one can see that the statistics of n-tuples anyons are described via the representations of braid groups.

References:

An Introduction to Quantum Computing for Non-Physicists (E. Rieffel, W. Polak), http://xxx.lanl.gov/PS_cache/quant-ph/pdf/9809/9809016.pdf

Topological quantum computation (Freedman, Kitaev, Larsen, Wang), Bull. AMS, vol 40 (2003), 31-38 http://arxiv.org/PS_cache/quant-ph/pdf/0101/0101025.pdf

Simulation of topological field theories by quantum computers (Freedman, Kitaev, Wang),  Commun. Math. Phys., vol 227 (2002), 587-60. http://arxiv.org/PS_cache/quant-ph/pdf/0001/0001071.pdf