Solution
= Solution
The <eigenphase Hadamard-test probability> is
$$
p_0=\frac{1+\operatorname{Re}e^{2\pi i\theta}}2
=\frac{1+\cos(2\pi\theta)}2
=\cos^2(\pi\theta).
$$
Thus
$$
\cos(2\pi\theta)=2p_0-1,
$$
or $\theta=\pm(2\pi)^{-1}\arccos(2p_0-1)\pmod1$. This test alone cannot distinguish $\theta$ from $1-\theta$.