A 3-Stranded Quantum Algorithm for the Jones Polynomial

Let K be a 3-stranded knot (or link), and let L denote the number of crossings in K. Let $ε_{1}$ and $ε_{2}$ be two positive real numbers such that $ε_{2}$ 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 $t=exp(iϕ)$ of the unit circle in the complex plane such that the absolute value of $ϕ$ is less than or equal to $π/3$. 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 $exp(iϕ))$ within a precision of $ε_{1}$ with a probability of success bounded below by $1-ε_{2}%. The execution time complexity of this algorithm is O(nL), where n is the ceiling function of (ln(4/ε_{2}))/(2(ε_{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).

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