Estimating Jones and HOMFLY polynomials with One Clean Qubit

TL;DR

This paper demonstrates that one clean qubit quantum computers can efficiently approximate Jones and HOMFLY polynomials at any root of unity, extending previous results limited to specific roots.

Contribution

It generalizes prior work by showing efficient approximation of Jones and HOMFLY polynomials at any root of unity using one clean qubit quantum computers.

Findings

Efficient approximation of Jones polynomial at any root of unity.

Extension of previous results to HOMFLY polynomial.

Demonstrates computational power of one clean qubit model.

Abstract

The Jones and HOMFLY polynomials are link invariants with close connections to quantum computing. It was recently shown that finding a certain approximation to the Jones polynomial of the trace closure of a braid at the fifth root of unity is a complete problem for the one clean qubit complexity class. This is the class of problems solvable in polynomial time on a quantum computer acting on an initial state in which one qubit is pure and the rest are maximally mixed. Here we generalize this result by showing that one clean qubit computers can efficiently approximate the Jones and single-variable HOMFLY polynomials of the trace closure of a braid at any root of unity.