  The optimal design of experiments typically involves solving an NP-hard
combinatorial optimization problem. In this paper, we aim to develop a globally
convergent and practically efficient optimization algorithm. Specifically, we
consider a setting where the pre-treatment outcome data is available and the
synthetic control estimator is invoked. The average treatment effect is
estimated via the difference between the weighted average outcomes of the
treated and control units, where the weights are learned from the observed
data. {Under this setting, we surprisingly observed that the optimal
experimental design problem could be reduced to a so-called \textit{phase
synchronization} problem.} We solve this problem via a normalized variant of
the generalized power method with spectral initialization. On the theoretical
side, we establish the first global optimality guarantee for experiment design
when pre-treatment data is sampled from certain data-generating processes.
Empirically, we conduct extensive experiments to demonstrate the effectiveness
of our method on both the US Bureau of Labor Statistics and the
Abadie-Diemond-Hainmueller California Smoking Data. In terms of the root mean
square error, our algorithm surpasses the random design by a large margin.
