Nuclear-magnetic-resonance quantum calculations of the Jones polynomial

Raimund Marx, Amr Fahmy, Louis Kauffman, Samuel Lomonaco, Andreas Spörl, Nikolas Pomplun, Thomas Schulte-Herbrüggen, John M. Myers, and Steffen J. Glaser
Phys. Rev. A 81, 032319 – Published 19 March 2010

Abstract

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.

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  • Received 10 September 2009

DOI:https://doi.org/10.1103/PhysRevA.81.032319

©2010 American Physical Society

Authors & Affiliations

Raimund Marx1, Amr Fahmy2,*, Louis Kauffman3, Samuel Lomonaco4, Andreas Spörl1, Nikolas Pomplun1, Thomas Schulte-Herbrüggen1, John M. Myers5, and Steffen J. Glaser1

  • 1Department of Chemistry, Technische Universität München, Lichtenbergstr. 4, D-85747 Garching, Germany
  • 2Biological Chemistry and Molecular Pharmacology, Harvard Medical School, 240 Longwood Avenue, Boston, Massachusetts 02115, USA
  • 3University of Illinois at Chicago, 851 S. Morgan Street, Chicago, Illinois 60607-7045, USA
  • 4University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, Maryland 21250, USA
  • 5Cruft Laboratory, Harvard University, 19 Oxford Street, Cambridge, Massachusetts 02138, USA

  • *To whom correspondence should be addressed: amr_fahmy@hms.harvard.edu

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Vol. 81, Iss. 3 — March 2010

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