Jones gives quantum algorithm for Jones knot polynomial

In summary, the conversation discusses the potential connection between quantum gravity and knots, with references to the work of Freedman, Kitaev, Larsen and Wang on TQFT and Aharonov et. al's paper on the polynomial quantum algorithm for approximating the Jones polynomial. Further research is needed to explore this connection.
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http://arxiv.org/abs/quant-ph/0511096
A Polynomial Quantum Algorithm for Approximating the Jones Polynomial
Dorit Aharonov, Vaughan Jones, Zeph Landau
26 pages

"The Jones polynmial, discovered in 1984, is an important knot invariant in topology, which is intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang provide an efficient simulation of TQFT by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient quantum algorithm that provides a certain additive approximation of the Jones polynomial at the fifth root of unity, and moreover, that this problem is BQP-complete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results of Freedman et. al are heavily based on deep knowledge of TQFT, which makes the algorithm essentially inaccessible for computer scientists.
We provide an explicit and simple polynomial algorithm to approximate the Jones polynomial of an n strands braid with m crossings at the primitive k'th root of unity, for any k, where the running time of the algorithm is polynomial in m,n and k. Our algorithm does not use TQFT at all. By the results of Freedman et. al, our algorithm solves a BQP complete problem.
The algorithm we provide exhibits a structure which we hope is generalizable to other quantum algorithmic problems. A candidate of particular interest is the approximation of the partition function of the Potts model."

John Baez is inviting grad students to join him at UC Riverside for research in QUANTUM MATHEMATICS, remarks Peter Woit

in some approaches to Quantum Gravity the quantum states of the gravitational field are knots---just an idle thought. does gravity, in other words quantum spacetime geometry, connect to this at all?
 
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That's an interesting thought! It's possible that the work of Freedman, Kitaev, Larsen and Wang could provide insight into the connection between quantum gravity and knots, since their work provides an efficient simulation of Topological Quantum Field Theory (TQFT). Aharonov et. al's paper on the polynomial quantum algorithm for approximating the Jones polynomial could also be relevant, as it provides a polynomial algorithm to approximate knot invariants. It would be interesting to see if further research can be done to explore this connection.
 
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I find this idea of using knots to study quantum gravity to be intriguing. While there is currently no definitive answer on the connection between quantum gravity and knot theory, there have been some interesting developments in this area. For example, the Jones polynomial has been used in the study of topological quantum field theory, which has implications for quantum gravity. Additionally, the concept of knot invariants has been applied in the context of quantum gravity, such as in the work of Louis Crane and Lee Smolin on the loop quantum gravity approach.

However, it is important to note that this is still an area of ongoing research and there is no clear consensus on the role of knot theory in quantum gravity. As a graduate student, joining John Baez at UC Riverside to explore this topic further could be a valuable opportunity to contribute to this exciting area of research. Ultimately, it is through continued investigation and collaboration that we can hope to uncover the potential connections between quantum gravity and knot theory.
 

1. What is the Jones knot polynomial?

The Jones knot polynomial is a mathematical tool used in knot theory to distinguish between different types of knots. It is a polynomial function that assigns a numerical value to each knot, allowing for easier identification and classification.

2. How does Jones' quantum algorithm work?

Jones' quantum algorithm is based on the principles of quantum computing, which uses quantum bits (qubits) instead of classical bits. It involves manipulating these qubits to perform calculations and solve problems related to the Jones knot polynomial.

3. What is the significance of Jones' algorithm for knot theory?

Jones' algorithm is a major breakthrough in knot theory, as it provides a more efficient and accurate method for calculating the Jones knot polynomial. It also has implications for other areas of mathematics and physics, such as topological quantum field theory.

4. How does Jones' algorithm compare to other methods for calculating the Jones knot polynomial?

Jones' algorithm is significantly faster and more efficient than previous methods for calculating the Jones knot polynomial. It also has the advantage of being applicable to a wider range of knots and can handle more complex calculations than other techniques.

5. Are there any practical applications for Jones' quantum algorithm?

While the immediate application of Jones' algorithm may be limited to theoretical mathematics and physics, it has the potential to contribute to the development of more powerful quantum computing technologies. It may also have applications in other fields such as cryptography and data encryption.

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