Topological quantum computing and the Jones polynomial

2006/04/29 by Samuel J. Lomonaco, Samuel J. Lomonaco, Jr., Louis H. Kauffman
Computer Science · Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algorithm #Combinatorics #Computer science #Discrete mathematics #Mathematical analysis #Mathematics #Physics #Polynomial #Quantum #Quantum algorithm #Quantum chaos and dynamical systems #Quantum computer #Quantum mechanics #Theoretical computer science #Time complexity #Topological and Geometric Data Analysis #Topology (electrical circuits) #quant-ph

paper · pdf · doi:10.1117/12.665361

19 pages, 27 figures

arxiv created 2006/04/29 · openalex publication_date 2006/05/05 · arxiv updated 2012/08/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this paper, we give a description of a recent quantum algorithm created by Aharonov, Jones, and Landau for approximating the values of the Jones polynomial at roots of unity of the form e^2πi/k. This description is given with two objectives in mind. The first is to describe the algorithm in such a way as to make explicit the underlying and inherent control structure. The second is to make this algorithm accessible to a larger audience.

Citations

• Quantum complexity theory
• On the computational complexity of the Jones and Tutte polynomials
• State models and the jones polynomial
• An Introduction to Knot Theory
• A Modular Functor Which is Universal¶for Quantum Computation
• Theory of Braids
• The Jones polynomial: quantum algorithms and applications in quantum complexity theory