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Nuclear-magnetic-resonance quantum calculations of the Jones polynomial

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posted on 03.12.2013, 00:00 by Raimund Marx, Amr Fahmy, Louis Kauffman, Samuel Lomonaco, Nikolas Pomplun, John M. Myers, Steffen J. Glaser, Andreas Spörl, Thomas Schulte-Herbrüggen
The repertoire of problems theoretically solvable by a quantum computer recently expanded to include the approximate evaluation of knot invariants, specifically the Jones polynomial. The experimental implementation of this evaluation, however, involves many known experimental challenges. Here we present experimental results for a small-scale approximate evaluation of the Jones polynomial by nuclear magnetic resonance (NMR); in addition, we show how to escape from the limitations of NMR approaches that employ pseudopure states. Specifically, we use two spin-1/2 nuclei of natural abundance chloroform and apply a sequence of unitary transforms representing the trefoil knot, the figure-eight knot, and the Borromean rings. After measuring the nuclear spin state of the molecule in each case, we are able to estimate the value of the Jones polynomial for each of the knots.


A.F. thanks NIH GM47467. S.G. acknowledges support from the integratedEUprojectQAP, the Deutsche Forschungsgemeinschaft (SFB-631), and from the Elite Network of Bavaria program QCCC.


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This is a copy of an article published in the Physical Review A © 2010 American Physical Society. The final publication is available at http://pra.aps.org/.


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