Professor Louis H. Kauffman
DEPARTMENT OF MATHEMATICS, STATISTICS, AND COMPUTER SCIENCE; UNIVERSITY OF ILLINOIS AT CHICAGO
Topological Quantum Information and the Jones Polynomial
We give a quantum statistical interpretation for the Jones polynomial in terms of the Kauffman bracket polynomial state sum. The Jones polynomial is a well-known topological invariant of knots in three-dimensional space that is closely related to structures in statistical mechanics and quantum field theory. We use this interpretation to give a new quantum algorithm for computing the Jones polynomial. This algorithm is useful for its conceptual simplicity, and it applies to all values of the polynomial variable that lie on the unit circle in the complex plane. Letting H(K) denote the Hilbert space for this model, there is a natural unitary transformation U from H(K) to itself such that = Trace(U) where is the bracket polynomial for the knot K. The quantum algorithm for arises directly from this formula via the Hadamard Test. We also review how we have implemented quantum algorithms for the Jones polynomial in NMR experiments and we show how the framework of the present model is related to recent work in knot theory such as Khovanov homology. This talk does not assume any background in either quantum computing or in the theory of knots and their invariants.