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 $N^{1/2}$.