Cross-posted on MO

The original ideas from the 70's/80's - that begat the [BB84] quantum key distribution - concerned quantum money that is unforgeable by virtue of the no-cloning theorem. A limitation was that the quantum money required the bank to verify each transaction. Quantum money based on knots and hidden subspaces have been explored to provide a "public key" quantum distribution.

I find the knots paper, aided by Farhi's exposition, to be quite accessible.

Enter the recent paper "Quantum Money from Modular Forms," by Daniel Kane.

For me modular forms are much more intimidating than knots; my knowledge is mostly limited to an excellent series of lectures by Keith Conrad and friends. However, Kane's exposition is very good and I can see some general relation to the previous work on knots.

Nonetheless, I'm getting hung up on section 3.2 onward. I'm wondering how much of a dictionary we can have between the knots work and modular forms.

That is, I know we can say something like there are $d!\times[\frac{d!}{e}]$ grid diagrams of grid dimension $d$, and a uniform superposition of grid diagrams all with the same Alexander polynomial is an eigenstate of a Markov chain of Cromwell moves; not only that, the Markov chain can be made doubly stochastic and easy to apply, and the Alexander polynomial is efficient to calculate.

Does it even make sense to say something roughly as in "there are $\lfloor N/12\rfloor$ cusp modular forms of weight $2$ and level $N$, and a uniform superposition of modular forms all with a same number of ideal classes is an eigenstate of a Hecke operator; not only that, the Hecke operator is Hermitian and easy to apply and the number of ideal classes is efficient to calculate?"

Have I gotten off track?

I think for me to make much more progress


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