I figured that a measurement of energy would collapse Farhi & Gutmann's analog analogue of Grover's algorithm to an energy eigenstate, which, when measured in the original basis, gives the marked state with probability 1/2. The problem is measuring the energy--since the energy levels are almost degenerate, coupling a pointer state with the energy in order to measure it either takes exponential time for the pointers to be distinguishable, or requires exponential energy to make them separate in constant time.

The paper.