How fast can a quantum state evolve into a target state?

Given a starting quantum state and a target quantum state, and a driving Hamiltonian, how fast can the starting state evolve into the target state according to the Schrödinger dynamics? This problem arises in a variety of contexts such as quantum computation, quantum control, and in particular, the problems of maximum information processing rate of quantum systems. We will establish a new time–energy uncertainty relation, which provides a general upper bound estimation of the evolution speed. A naive yet fundamental example which has interesting implications for quantum search is worked out explicitly.