In addition to that, the absolutely enormous domains that the Fourier Transform sums over (essentially, one term in the sum for each possible answer), and the cancellations which would have to occur for that sum to be informative, means that a theoretically-capable Quantum Computer will be testing the predictions of Quantum Mechanics to a degree of precision hundreds of orders of magnitude greater than any physics experiment to date. (Or at least dozens of orders of magnitude, in the case of breaking Discrete Log on an Elliptic Curve.) It demands higher accuracy in the probability distributions predicted by QM than could be confirmed by naive frequency tests which used the entire lifetime of the entire universe as their laboratory!

Imagine a device conceived in the 17th century, the intended functionality of which would require a physical sphere which matches a perfect, ideal, geometric sphere in Euclidean space to thousands of digits of precision. We now know that the concept of such a perfect physical sphere is incoherent with modern physics in a variety of ways (e.g., atomic basis of matter, background gravitational waves.) I strongly suspect that the cancellations required for the Fourier Transform in Shor's algorithm to be cryptographically relevant will turn out to be the moral equivalent of that perfect sphere.

We'll probably learn some new physics in the process of trying to build a Quantum Computer, but I highly doubt that we'll learn each others' secrets.