# Existence of unitary matrix that produces quantum coin with given frequencies via iteration

I'm curious about for which $$n$$ there exists a $$2\times 2$$ unitary matrix $$U(n)$$ such that for $$1 \le k \le n$$ $$|U^k_{1,0}(n)|^2 = \frac kn,$$ where $$U^k_{1,0}(n)$$ is the lower left element of the $$k$$th power of $$U(n)$$.

The idea is that if such a matrix exists, then applying it $$k$$ times to the quantum state $$|0\rangle$$ will produce the quantum state $$\sqrt{1-k/n}|0\rangle + \sqrt{k/n}|1\rangle$$ modulo some phases, a "quantum coin" with probability $$k/n$$ of giving outcome 1.

It's easy to see that it exists for $$n=2$$, with an example being $$U(2) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix},$$ but I couldn't find an example for any other $$n$$. A bit of brute force shows that no such matrices exist for $$n=3$$, and suggests that indeed no other $$n$$ works. Perhaps there's an elegant way to prove this? Or maybe there is a construction that I missed?