Approximating the Jones polynomial: an NMR experiment
Coauthors: O. Moussa, C.A. Ryan, R. Laflamme
The class of problems efficiently solvable on quantum computers with one bit of quantum information is known as DQC1 [1]. This model is believed to be strictly weaker than standard quantum computers, but still more powerful than their classical counterparts. Recently, Shor and Jordan [2] proved that the problem of approximating the Jones polynomial at the fifth root of unity completely encapsulates the power of DQC1. The Jones polynomial is a knot invariant that is not only important to knot theory, but also to statistical mechanics and quantum field theory. We present an adaptation of the algorithm developed by Shor and Jordan suitable for implementation on a liquid state NMR quantum information processor, and report on the experimental implementation of the algorithm to evaluate the Jones polynomial for all knots whose braid representation has four strands and three crossings.
References: [1] E. Knill and R. Laflamme. Phys. Rev. Lett., 81, 5672 (1998).
[2] P. Shor and S. Jordan. Quant. Inf. and Comm., 8, 681-714, (2008).