# Topological quantum computing and the Jones polynomial

@inproceedings{Lomonaco2006TopologicalQC, title={Topological quantum computing and the Jones polynomial}, author={Samuel J. Lomonaco and Louis H. Kauffman}, booktitle={SPIE Defense + Commercial Sensing}, year={2006} }

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 e2πi/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.

#### 32 Citations

A quantum manual for computing the Jones polynomial

- Mathematics, Engineering
- SPIE Defense + Commercial Sensing
- 2008

The objective of this paper is to give experimentalists a quantum manual for implementing the Aharonov-Jones- Landau algorithm on an architecture of their choice. In particular, we explicitly apply… Expand

Quantum knitting

- Physics
- 2006

We analyze the connections between the mathematical theory of knots and quantum physics by addressing a number of algorithmic questions related to both knots and braid groups. Knots can be… Expand

The Jones polynomial: quantum algorithms and applications in quantum complexity theory

- Computer Science, Physics
- Quantum Inf. Comput.
- 2008

It is concluded with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the TuttePolynomial and graph coloring. Expand

Gravitational Topological Quantum Computation

- Mathematics, Computer Science
- UC
- 2007

Some applications of GTQC in quantum complexity theory and computability theory are discussed, particularly it is conjectured that the Khovanov polynomial for knots and links is more hard than #P-hard; and that the homeomorphism problem, which is noncomputable, maybe can be computed after all via a hyper-computer based onGTQC. Expand

$q$ - Deformed Spin Networks, Knot Polynomials and Anyonic Topological Quantum Computation

- Mathematics, Physics
- 2006

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups.… Expand

A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

- Mathematics, Computer Science
- Algorithmica
- 2008

An explicit and simplePolynomial quantum algorithm to approximate the Jones polynomial of an n strands braid with m crossings at any primitive root of unity e2πi/k, where the running time of the algorithm is polynometric in m, n and k. Expand

Quantum Algorithms For: Quantum Phase Estimation, Approximation Of The Tutte Polynomial And Black-box Structures

- Mathematics
- 2012

In this dissertation, we investigate three different problems in the field of Quantum computation. First, we discuss the quantum complexity of evaluating the Tutte polynomial of a planar graph.… Expand

Introduction to topological quantum computation with non-Abelian anyons

- Computer Science, Physics
- 2018

This work aims to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Expand

Quantum Algorithms

- Computer Science, Physics
- Encyclopedia of Complexity and Systems Science
- 2009

This work provides an explicit algorithm for generating any prescribed interference pattern with an arbitrary precision for quantum algorithms and shows how they are related to diierent instances of quantum phase estimation. Expand

Efficient quantum processing of three–manifold topological invariants

- Physics, Mathematics
- 2009

A quantum algorithm for approximating efficiently three–manifold topological invariants in the framework of SU(2) Chern–Simons–Witten (CSW) topological quantum field theory at finite values of the… Expand

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The Jones polynomial: quantum algorithms and applications in quantum complexity theory

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It is concluded with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the TuttePolynomial and graph coloring. Expand

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