Approximating Jones Polynomial in Knot Theory using AJL Algorithm

[ucantfindme]

Proposal

Team - Bell Basis

• Abhijeet Bhatta (@AbhijeetBhatta-DiracDeltaDev) - M.Tech Quantum Technology, IISc Bangalore
• Shubham Sharma (@shubham172995) - M.Tech Quantum Technology, IISc Bangalore
• Sai Pavan Krishna (@ucantfindme) - M.Tech Quantum Technology, IISc Bangalore
• Ritvik Raina (@RitvikRaina) - M.Tech Quantum Technology, IISc Bangalore

Paper Details

Title: A Polynomial Quantum Algorithm for Approximating the Jones Polynomial
Authors: Dorit Aharonov, Vaughan Jones, Zeph Landau

Brief Problem Statement

The Jones polynomial, a knot invariant, is deeply linked to Topological Quantum Field Theory (TQFT) and quantum computing. Prior research suggested an efficient quantum algorithm for approximating it at the fifth root of unity, but no explicit formulation was available. This paper presents a polynomial-time quantum algorithm for approximating the Jones polynomial at any primitive root of unity, avoiding TQFT. The approach has potential generalization to #P-hard problems such as the Potts model.

Implementation Approach

1. Braid Representation: Every knot is represented as either a plat-closure or a trace-closure of a braid.
2. Unitary Operator Construction: A unitary matrix is derived from the braid word of the given braid.
3. Quantum Computation Using the Hadamard Test:
   • The Hadamard Test is applied to obtain the real and imaginary components of the expected value of the unitary matrix.
   • This test is repeated a polynomial number of times proportional to the number of crossings in the braid.
4. Classical Computation: The weight of the braids is computed in polynomial time using classical methods.
5. Jones Polynomial Calculation: Using the computed parameters, the Jones Polynomial is evaluated at a root of unity.

Simple Illustration

The image illustrates three fundamental knots in knot theory: the Hopf Link, Trefoil Knot, and Figure-Eight Knot. Their corresponding braid representations and projections highlight their topological significance in computing invariants such as the Jones polynomial.

This table extends the previous image by linking knots to their corresponding braid words and diagrams. It includes the Hopf Link, Trefoil Knot, Figure-Eight Knot, and Whitehead Link, with braid group notation (e.g., ( \tau_1^3 ) for the Trefoil Knot). These knots are crucial in topology and quantum computing, particularly in computing invariants like the Jones polynomial.

Future Directions

• Knot Classification via Jones Polynomial Approximation: Convert knots into braid closures and approximate the Jones polynomial at different roots of unity. If two knots yield different polynomials, they must be distinct. This method enables a database-driven classification of knots.

Real-World Applications

1. Protein Folding:

   • Protein knotting relates to knot theory and the Jones polynomial, assisting in classifying folding pathways.
   • Just as the polynomial distinguishes knots, topology aids in predicting protein structures.
   • Analyzing crystal structures helps validate Taylor’s twisted hairpin model and suggests alternative folding mechanisms.
   • This enhances our understanding of protein function and evolution.

2. Knot Theory in Statistical Physics:

   • The Jones polynomial and the Potts model connect through the Tutte polynomial, linking knot theory and statistical mechanics.
   • Efficient computation of the Jones polynomial would improve:
     • Knot classification
     • Topological quantum computing
     • Phase transition analysis in physics
   • These insights could also optimize algorithms in quantum simulations and graph theory applications.

References

• The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots
• The Jones Polynomial
• Topological Descriptions of Protein Folding

[NadavClassiq]

Sounds like an interesting project, @ucantfindme !

We would be glad to have such a contribution in our repository!

Please note that we accept high-quality implementations in our repository and will be glad to accept a contribution that meets our standards.

Given that the Classiq platform focuses on high-level quantum circuit synthesis, could you clarify how you plan to implement the AJL algorithm within this framework?

Feel free to reach out to the community if you have any questions.

Good luck!