I was wondering if we can setup a superposition of the elements of a finite field $\mathbb{F}_p$ with $p \in \mathbb{P}$ in polynomial time. I am only interested in the prime case, but if you want to generalise to $p^k$ feel free.
i.e. I am interested having one of these two superpositions: $$ \frac{1}{\sqrt{p}} \sum_{i=0}^{p-1}|i\rangle \text{ and } \frac{1}{\sqrt{p-1}} \sum_{i=1}^{p-1}|i\rangle $$ and if they can be constructed in polynomial time. I believe there is an approach that takes advantage of Grover's search as highlighted here: https://quantumcomputing.stackexchange.com/a/15528/30140, however I want to see if I overlooked something more basic.
