# A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

@article{Aharonov2006APQ, title={A Polynomial Quantum Algorithm for Approximating the Jones Polynomial}, author={Dorit Aharonov and Vaughan F. R. Jones and Zeph Landau}, journal={Algorithmica}, year={2006}, volume={55}, pages={395-421} }

AbstractThe Jones polynomial, discovered in 1984, is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten) to be intimately connected to Topological Quantum Field Theory (
${\sf{TQFT}}$
). The works of Freedman, Kitaev, Larsen and Wang provide an efficient simulation of
${\sf{TQFT}}$
by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient (namely, polynomial…

## 205 Citations

Efficient quantum circuits for approximating the Jones polynomial

- Computer ScienceQuantum Inf. Comput.
- 2008

A new method is proposed for implementing the AJL algorithm, which improves the performance from O (mnlog2 k) to O(mn), where n is the number of strands, m is thenumber of the crossings in a braid and k is a large-degree polynomial.

Quantum Algorithms Beyond the Jones Polynomial

- Mathematics, Computer Science
- 2007

An additive approximation of the partition function of the Potts model with any set of couplings at any temperature is achieved, at any point, for any planar graph.

Eciently Computing the Jones Polynomial on a Quantum Computer

- Computer Science
- 2015

The aim of these notes is to present their algorithm with enough background information so that it can be understood by someone who does not have a background in knot theory or quantum computing.

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

- Mathematics, Computer ScienceQuantum 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.

Topological Quantum Information, Khovanov Homology and the Jones Polynomial

- Mathematics
- 2010

In this paper we give a quantum statistical interpretation for the bracket polynomial state sum and for the Jones polynomial. We use this quantum mechanical interpretation to give a new quantum…

How Hard Is It to Approximate the Jones Polynomial?

- MathematicsTheory Comput.
- 2015

Any value-dependent approximation of the Jones polynomial at these non-lattice roots of unity is #P-hard, which follows fairly directly from the universality result and Aaronson's theorem that PostBQP = PP.

Quantum algorithms for virtual Jones polynomials via Thistlethwaite theorems

- Computer ScienceDefense + Commercial Sensing
- 2010

A quantum algorithm for the Jones polynomial of a given virtual link in terms of the generalized Tutte polynomials by exploiting the Thistlethwaite theorem and the Kauffman algorithm is claimed as the quantum version of the Diao-Hetyei method.

Efficient quantum processing of 3-manifold topological invariants

- Physics
- 2007

All the significant quantities - partition functions and observables - of quantum CSW theory can be processed efficiently on a quantum computer, reflecting the intrinsic, field-theoretic solvability of such theory at finite k.

Polynomial Quantum Algorithms for Additive approximations of the Potts model and other Points of the Tutte Plane

- Computer Science, Mathematics
- 2007

The main progress in this work is in the ability to handle non-unitary representations of the Temperley Lieb algebra, both when applying them in the algorithm, and in the proof of universality, when encoding quantum circuits using non- unitary operators.

Quantum automata, braid group and link polynomials

- Computer ScienceQuantum Inf. Comput.
- 2007

In this combinatorial framework, families of finite-states and discrete-time quantum automata capable of accepting the language generated by the braid group are implemented, whose transition amplitudes are colored Jones polynomials.

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