A 3-Stranded Quantum Algorithm for the Jones Polynomial

TL;DR

This paper introduces two algorithms, classical and quantum, for efficiently computing the Jones polynomial of 3-stranded knots at specific points on the unit circle, with the quantum version offering probabilistic estimates.

Contribution

It presents a novel quantum algorithm for approximating the Jones polynomial of 3-stranded knots with improved time complexity over classical methods.

Findings

Classical algorithm runs in linear time O(L).

Quantum algorithm estimates the Jones polynomial with probabilistic accuracy.

Quantum algorithm's runtime is O(nL), with n depending on desired precision and success probability.

Abstract

Let K be a 3-stranded knot (or link), and let L denote the number of crossings in K. Let $\epsilon_{1}$ and $\epsilon_{2}$ be two positive real numbers such that $\epsilon_{2}$ is less than or equal to 1. In this paper, we create two algorithms for computing the value of the Jones polynomial of K at all points $t=exp(i\phi)$ of the unit circle in the complex plane such that the absolute value of $\phi$ is less than or equal to $\pi/3$. The first algorithm, called the classical 3-stranded braid (3-SB) algorithm, is a classical deterministic algorithm that has time complexity O(L). The second, called the quantum 3-SB algorithm, is a quantum algorithm that computes an estimate of the Jones polynomial of K at $exp(i\phi))$ within a precision of $\epsilon_{1}$ with a probability of success bounded below by $1-\epsilon_{2}%. The execution time complexity of this algorithm is O(nL), where n is the ceiling function of (ln(4/\epsilon_{2}))/(2(\epsilon_{2})^2). The compilation time complexity, i.e., an asymptotic measure of the amount of time to assemble the hardware that executes the algorithm, is O(L).