Experimental approximation of the Jones polynomial with one quantum bit

We present experimental results approximating the Jones polynomial using 4 qubits in a liquid state nuclear magnetic resonance quantum information processor. This is the first experimental implementation of a complete problem for the DQC1 model of quantum computation, which uses a single qubit accompanied by a register of completely random states. The Jones polynomial is a knot invariant that is important not only to knot theory, but also to statistical mechanics and quantum field theory. The implemented algorithm is a modification of the algorithm developed by Shor and Jordan suitable for implementation in NMR. These experimental results show that for the restricted case of knots whose braid representations have four strands and exactly three crossings, identifying distinct knots is possible 91 % of the time. These results demonstrate the high level of experimental control currently available in liquid state NMR. PACS numbers: 03.67.Lx, 76.60.-k. Quantum information processors have the potential to solve some problems exponentially faster than current classical methods [1]. While much effort has been concen-trated on the most conventional circuit model of compu