The Tale of One-Way Functions (section 2.4) claims that quantum mechanics would have to be accurate to an remarkable (absurd?) degree for an application such as factoring large numbers. So, orthogonal to the thousands of qubits and billions of gates that might be necessary to break 2048-bit RSA, can someone give a numerical estimate of this? Something along the lines of
Factoring $N$-bit numbers with Shor would require $D$ digits of accuracy in qubit amplitudes.
More generally, is this something that people (you quantum computer people) consider, i.e. models of quantum computing with bounded precision amplitudes?
(Please forgive my ignorance and inability/unwillingness to go through the algorithm and come up with an answer myself.)
Addendum:
I suppose what I had in mind is along the lines of simulating a quantum computer with a classical computer, with some finite representation of $\mathbb{C}^2$ for qubits. If one does all the exponentiations, DFTs, sums over the superpositions, etc., would the necessary size of that "finite representation" be ludicrous in order to achieve the necessary interference?
However, this mindset may not be physical or model-independent, and might not give any intuition about quantum computing in any case.
Is there some way of quantifying a regime where we experimentally feel that linearity holds and can this be translated into parameters going outside that regime for something "practical" like factoring? Or, at what point does something like factoring start testing quantum mechanics?