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 V_K(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 V_K (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).

Date Published: 10 May 2007
PDF: 18 pages
Proc. SPIE 6573, Quantum Information and Computation V, 65730T (10 May 2007); doi: 10.1117/12.719399

Published in SPIE Proceedings Vol. 6573:
Quantum Information and Computation V
Eric J. Donkor; Andrew R. Pirich; Howard E. Brandt, Editor(s)