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UNTYING KNOTS BY NMR: EXPERIMENTAL IMPLEMENTATION OF AN EXPONENTIALLY FAST QUANTUM ALGORITHM FOR APPROXIMATING THE JONES POLYNOMIAL

Raimund Marx

Published Dec 1, 2012• 0 citations
Publisher
DOI
10.1142/9789814425988_0002

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Abstract. The permanent of a square matrix is defined in a way similar to the determinant, but without using signs. The exact computation of the permanent is hard, but there are Monte-Carlo algorithms that can estimate general permanents. Given a planar diagram of a link L with n crossings, we define a 7n×7n matrix whose permanent equals to the Jones polynomial of L. This result accompanied with recent work of Freedman, Kitaev, Larson and Wang ([FKLW]) provides a Monte-Carlo algorithm to any decision problem belonging to the class BQP, i.e. such that it can be computed with bounded error in polynomial time using quantum resources. 1.
Theoretical Quantum Information Processing (QIP) has matured from the use of qubits to the use of qudits (systems having states >2). Whereas most of the experimental implementations have been performed using qubits, little experimental work has been carried out using qudits as yet. In this paper we demonstrate experimental realization of a qutrit system by nuclear magnetic resonance (NMR), utilizing deuterium (spin-1) nuclei partially oriented in liquid crystalline phase. Preparation of pseudopure states and implementation of unitary operations are demonstrated in this single-qutrit system, using transition selective pulses.