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 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.
Highlights
The Jones polynomial [1], a great discovery in knot theory, has recently become an interesting topic for quantum computing
In [3, 4] a quantum algorithm is given by Kauffman and Lomonanco (KL) for three-strand braids that can be used to evaluate the Jones polynomial at a continous range of angles
The method of estimation described for the AJL algorithm and the KL algorithm requires that the quan
Summary
The Jones polynomial [1], a great discovery in knot theory, has recently become an interesting topic for quantum computing. The use of quantum computing has been discussed for approximately evaluating the Jones polynomial V (z) at selected values of z. For a knot displayed as a braid of n strands (specified in terms of a sequence of crossings), these are the values z of the form z = exp(2πi/k) where k is an integer in the algorithm of Aharonov, Jones and Landau (AJL) [2]. In [3, 4] a quantum algorithm is given by Kauffman and Lomonanco (KL) for three-strand braids that can be used to evaluate the Jones polynomial at a continous range of angles. Most of the computational cost of the approximate evaluation is the estimation of the trace of a unitary matrix. The method of estimation described for the AJL algorithm and the KL algorithm requires that the quan-
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