A 3-stranded quantum algorithm for the Jones Polynomial
Louis Hirsch Kauffman, Samuel J. Lomonaco · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 2007
Let K be a 3-stranded knot (or link), and let L denote the number of crossings in of K. Let ε1 and ε2 be two positive real numbers such that ε2≤1. In this paper, we create two algorithms for computing the value of the Jones polynomial VK(t) at all points t=exp(iφ) of the unit circle in the complex plane such that |φ|≤2π/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 VK (exp (iφ)) within a precision of ε1 with a probability of success bounded below by 1-ε2. The execution time complexity of this algorithm is O(nL), where n is the ceiling function of (ln(4/ε2))/2ε21 . 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).