What can be said about the eigenvalues of a unitary quantum gate $U$ which implements $f(x)=f(x+r) \forall x$, as in $U|x\rangle|0\rangle\rightarrow|x\rangle|f(x)\rangle$ ?

The reason I'm asking is that in phase estimation and other related algorithms, all the eigenvalues are of the form $e^{2\pi i (s/r)}, s \in {0,1,2,\dots,r-1}$. Is this a general phenomenon?

  • $\begingroup$ Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. $\endgroup$
    – Community Bot
    Commented Nov 16, 2023 at 6:35
  • 4
    $\begingroup$ We need to know more generally how $U$ works. Is it true that $U|x\rangle|y\rangle=|x\rangle|y\oplus f(x)\rangle$? (In which case, $U^2=I$, so the eigenavlues are $\pm 1$.) $\endgroup$
    – DaftWullie
    Commented Nov 16, 2023 at 8:07