  We study the performance of the spectral method for the phase synchronization
problem with additive Gaussian noises and incomplete data. The spectral method
utilizes the leading eigenvector of the data matrix followed by a normalization
step. We prove that it achieves the minimax lower bound of the problem with a
matching leading constant under a squared $\ell_2$ loss. This shows that the
spectral method has the same performance as more sophisticated procedures
including maximum likelihood estimation, generalized power method, and
semidefinite programming, as long as consistent parameter estimation is
possible. To establish our result, we first have a novel choice of the
population eigenvector, which enables us to establish the exact recovery of the
spectral method when there is no additive noise. We then develop a new
perturbation analysis toolkit for the leading eigenvector and show it can be
well-approximated by its first-order approximation with a small $\ell_2$ error.
We further extend our analysis to establish the exact minimax optimality of the
spectral method for the orthogonal group synchronization.
