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Optimization of constrained quantum control problems powers quantum technologies. This task becomes very difficult when these control problems are non-convex and plagued with dense local extrema. For such problems current optimization methods must be repeated many times to find good solutions, each time requiring many simulations of the system. Here, we present Quantum Control via Polynomial Optimization (QCPOP), a method that eliminates this problem by directly finding globally optimal solutions. The resulting increase in speed, which can be a thousandfold or more, makes it possible to solve problems that were previously intractable. This remarkable advance is due to global optimization methods recently developed for polynomial functions. We demonstrate the power of this method by showing that it obtains an optimal solution in a single run for a problem in which local extrema are so dense that gradient methods require thousands of runs to reach a similar fidelity. Since QCPOP is able to find the global optimum for quantum control, we expect that it will not only enhance the utility of quantum control by making it much easier to find the necessary protocols, but provide a key tool for understanding the precise limits of quantum technologies. Finally, we note that the ability to cast quantum control as polynomial optimization resolves an open question regarding the computability of exact solutions to quantum control problems.