I dropped QM from my chemistry classes because of this and did laserspectroscopy PhD instead, but the basic principle of uncertainty (non-)measurement in QM is very similar to determining the life/death of Schroedingers cat experiment. Bear with me for i am just a physical-chemist, not hardcore like Sandre
Put a cat in a light and soundsealed box, place a contraption with cyanide gas inside, which is triggered by a photon, and place in front of the contraption a 50% transparent mirror, shoot a photon at it and start discussing if the cat is alive or dead.
Naturally the photon will either reflect (cat ia alive) or pass through (cat is dead). But the trick is that it is NOT not known if the cat is alive or dead, but that it cannot be determined, because you sealed the box.
The cat is now in a 50% dead 50% alive situation, according to QM theory. Exactly this "superposition" is crucial with respect to the quantum experiments based on solving algorithms.
When running such an experiment, all solutions of the algorithm appear quickly and on top of eachother (wavefunction superposition), and when you keep the box closed, like in Schroedingers cat, all possible outcomes of the algorithm are intact.
As soon as you open the box and start measuring, the solutions collapse into only one state, the "solution" you just measured, and you've ruined the algorithm calculation.
So the trick is to "read" the solution without looking into the box. I think this has been achieved by Kwiat. How exactly he did it, i don't know, because the counterfactual experiment baffles me. By not running it, so not letting the photons do their work, he found the solution?
I would rather see something like, by not running through MC you complete MC? I have hopes for my Pally