Analog analogue of a digital quantum computation

We solve a problem, which while not fitting into the usual paradigm, can be viewed as a quantum computation. Suppose we are given a quantum system with a Hamiltonian of the form E|w〉〈w| where |w〉 is an unknown (normalized) state. The problem is to produce |w〉 by adding a Hamiltonian (independent of |w〉) and evolving the system. If |w〉 is chosen uniformly at random we can (with high probability) produce |w〉 in a time proportional to N^1/2/E. If |w〉 is instead chosen from a fixed, known orthonormal basis we can also produce |w〉 in a time proportional to N^1/2/E and we show that this time is optimally short. This restricted problem is an analog analogue to Grover's algorithm, a computation on a conventional (!) quantum computer that locates a marked item from an unsorted list of N items in a number of steps proportional to N^1/2.