# A 3-Stranded Quantum Algorithm for the Jones Polynomial - Quantum Physics

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Abstract: Let K be a 3-stranded knot or link, and let L denote the number ofcrossings in K. Let $\epsilon {1}$ and $\epsilon {2}$ be two positive realnumbers such that $\epsilon {2}$ is less than or equal to 1.In this paper, we create two algorithms for computing the value of the Jonespolynomial of K at all points $t=expi\phi$ of the unit circle in the complexplane 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 OL. Thesecond, called the quantum 3-SB algorithm, is a quantum algorithm that computesan estimate of the Jones polynomial of K at $expi\phi$ 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 OnL, where n is theceiling function of ln4-\epsilon {2}-2\epsilon {2}^2. The compilationtime complexity, i.e., an asymptotic measure of the amount of time to assemblethe hardware that executes the algorithm, is OL.

Autor: Louis H. Kauffman, Samuel J. Lomonaco, Jr

Fuente: https://arxiv.org/