Period finding for Quantum Computers
Can quantum computer find the period of any given function efficiently? Are there any requirements towards the function?

Well, the function has to be periodic; that is, we have, for some $c &gt; 0$, $f(x) = f(x+c)$, for all $x$ [1].
For both factoring and discrete log, we have such a function; and we know how finding the period of that function will give us the answer.
However, postquantum algorithms, we don't have such a way.  For code-based schemes, no periodicity is apparent.  For lattice-based schemes, there is (periodicity is inherent in the idea of a lattice), however we don't know how to convert finding such a periodicity into (say) an answer to the shortest vector problem.

[1]: Actually, I believe Shor's algorithm will work even if the above relation holds for most $x$; however for the functions of interest, true periodicity holds...