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.