  We introduce an efficient iterative method to prepare a target state in
Hilbert spaces with high dimensionality using a combination of unitary
evolution, measurements, and quantum Zeno dynamics. The latter confines the
evolution within Zeno subspaces of decreasing size. This gives an exponential
speed up relative to the case of states evolving in the full Hilbert space
between projective measurements. We demonstrate our approach on the control
problem of rapidly transferring a superfluid into the Mott insulator in the
Bose-Hubbard model. We discuss the general applicability of the method by
preparing arbitrary superpositions with random Hamiltonians.
