Experimental approximation of the Jones polynomial with DQC1

TL;DR

This paper demonstrates the first experimental approximation of the Jones polynomial using a DQC1 quantum computer with 4 qubits, achieving 91% accuracy for specific knots, showcasing potential for quantum knot invariants.

Contribution

It provides the first experimental implementation of a complete DQC1 algorithm for approximating the Jones polynomial using liquid state NMR quantum computing.

Findings

Achieved 91% success rate in distinguishing certain knots.

Implemented a modified DQC1 algorithm suitable for NMR systems.

First experimental realization of a complete DQC1 problem.

Abstract

We present experimental results approximating the Jones polynomial using 4 qubits in a liquid state nuclear magnetic resonance quantum information processor. This is the first experimental implementation of a complete problem for the deterministic quantum computation with one quantum bit model of quantum computation, which uses a single qubit accompanied by a register of completely random states. The Jones polynomial is a knot invariant that is important not only to knot theory, but also to statistical mechanics and quantum field theory. The implemented algorithm is a modification of the algorithm developed by Shor and Jordan suitable for implementation in NMR. These experimental results show that for the restricted case of knots whose braid representations have four strands and exactly three crossings, identifying distinct knots is possible 91% of the time.