  We study the phase synchronization problem with measurements
$Y=z^*z^{*H}+\sigma W\in\mathbb{C}^{n\times n}$, where $z^*$ is an
$n$-dimensional complex unit-modulus vector and $W$ is a complex-valued
Gaussian random matrix. It is assumed that each entry $Y_{jk}$ is observed with
probability $p$. We prove that the minimax lower bound of estimating $z^*$
under the squared $\ell_2$ loss is $(1-o(1))\frac{\sigma^2}{2p}$. We also show
that both generalized power method and maximum likelihood estimator achieve the
error bound $(1+o(1))\frac{\sigma^2}{2p}$. Thus, $\frac{\sigma^2}{2p}$ is the
exact asymptotic minimax error of the problem. Our upper bound analysis
involves a precise characterization of the statistical property of the power
iteration. The lower bound is derived through an application of van Trees'
inequality.
