Citation: Farhi, Edward, et al. "Quantum money from knots." Proceedings of the 3rd Innovations in Theoretical Computer Science Conference. 2012.
Web: https://arxiv.org/abs/1004.5127
Tags: Computer-scientific, Philosophical
In this paper, the authors propose a scheme for quantum money based on knot theory. The idea is to use the hardness of distinguishing knots to create the scheme.
The scheme works like this. The mint creates a state which is the weighted superposition over knots with a given Alexander polynomial. This is easy to do by first creating a superposition over knots, and then measuring the Alexander polynomial. Creating the state which is superposition over knots with the same specific given Alexander polynomial is hard though. Hence, it might make for good quantum money.
This paper seems like it was well received, but was quickly superseded by Aaronson and Christiano's quantum money scheme:
> Aaronson, Scott, and Paul Christiano. "Quantum money from hidden subspaces." Proceedings of the forty-fourth annual ACM symposium on Theory of computing. 2012.
A good popular review of quantum money is here:
> Aaronson, Scott, et al. "Quantum money." Communications of the ACM 55.8 (2012): 84-92.
My take on this paper is that it is another nice example of the relationship between topology and quantum information. Proving hardness results like "approximating Jones polynomial = BQP-complete" seem like they may be useful for certain more pure quantum-information applications, like this one. Any proof that the quantum money scheme was secure would necessarily be based in some sort of quantum topology computational complexity hardness result.