A 3-stranded quantum algorithm for the Jones Polynomial

Louis H. Kauffman; Samuel J. Lomonaco Jr.

doi:10.1117/12.719399

10 May 2007 A 3-stranded quantum algorithm for the Jones Polynomial

Louis H. Kauffman, Samuel J. Lomonaco Jr.

Proceedings Volume 6573, Quantum Information and Computation V; 65730T (2007) https://doi.org/10.1117/12.719399
Event: Defense and Security Symposium, 2007, Orlando, Florida, United States

Abstract

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).

Citation Download Citation

Louis H. Kauffman and Samuel J. Lomonaco Jr. "A 3-stranded quantum algorithm for the Jones Polynomial", Proc. SPIE 6573, Quantum Information and Computation V, 65730T (10 May 2007); https://doi.org/10.1117/12.719399

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