Abstract for Lecture II:

In this lecture, I will describe the Jones representations of the braid groups, and use it to build topological quantum computers. Then I will answer the question: when are the resulting quantum computers universal? The Jones representation of the braid groups are indexed by an integer r. It turns out that unless r=1,2,3,4,6, then we have a universal quantum computer.

Statistics of anyons are described by the representations of the braid groups. One of the most interesting representations of the braid groups is Jones’s unitary representation arising from subfactor theory in the theory of operator algebras. The discovery of this representation led Jones to his celebrated polynomial invariant of knots and to the ongoing quantum revolution in topology. It turns out the computational power of topological quantum computers depends on the size of the closed images of the Jones representation in the unitary groups.

In studying universality for quantum computing, we also answered a question of Jones:

A pair (G,V) consists of a compact, not necessarily connected, Lie group G, a finite dimensional Hilbert space V, and a faithful irreducible complex representation R of G into the group U(V) of unitary automorphisms of V. We say a pair (G,V) satisfies the 2-eigenvalue property if there exists a g in G such that (i) the conjugacy class of g generates G topologically, and (ii) the eigenvalue spectrum of R(g) consists of exactly 2 numbers whose ratio is not -1.

The problem of Jones was to classify pairs (G, V) having the 2-eigenvalue property. The method employed adapts also to the 3-eigenvalue problem, which has been solved for continuous Lie groups, but it remains open for finite groups. For N>3, the problem is completely open.

References for Lecture 2:

A modular functor which is universal for quantum computation (Freedman, Larsen, Wang), Commun. Math. Phys., vol 227 (2002), 605-62. http://arxiv.org/PS_cache/quant-ph/pdf/0001/0001108.pdf

The two-eigenvalue problem and density of Jones, representation of braid groups,  (Freedman, Larsen, Wang), Commun. Math. Phys., vol 228 (2002),177-199. http://arxiv.org/PS_cache/math/pdf/0103/0103200.pdf