NMR Quantum Calculations of the Jones Polynomial

TL;DR

This paper demonstrates a small-scale experimental approach using NMR to approximate the Jones polynomial of knots, addressing challenges of NMR methods and showcasing practical quantum knot invariant evaluation.

Contribution

It presents the first NMR-based experimental approximation of the Jones polynomial for specific knots, overcoming limitations of pseudo pure states in NMR quantum computing.

Findings

Successfully estimated Jones polynomial values for three knots

Used NMR with natural abundance chloroform nuclei

Implemented unitary transforms representing knots

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 pseudo pure 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 state of the molecule in each case, we are able to estimate the value of the Jones Polynomial for each of the knots.