  We study the phase synchronization problem with noisy 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 an SDP relaxation of the MLE achieves the error
bound $(1+o(1))\frac{\sigma^2}{2np}$ under a normalized squared $\ell_2$ loss.
This result matches the minimax lower bound of the problem, and even the
leading constant is sharp. The analysis of the SDP is based on an equivalent
non-convex programming whose solution can be characterized as a fixed point of
the generalized power iteration lifted to a higher dimensional space. This
viewpoint unifies the proofs of the statistical optimality of three different
methods: MLE, SDP, and generalized power method. The technique is also applied
to the analysis of the SDP for $\mathbb{Z}_2$ synchronization, and we achieve
the minimax optimal error $\exp\left(-(1-o(1))\frac{np}{2\sigma^2}\right)$ with
a sharp constant in the exponent.
