Quantum mechanics, as we all know, is weird. It's weird enough in its own right, but when some determined experimenters do tricks that really bring out the weirdness in all its glory, and the results are conveyed to us by well-intentioned but occasionally murky vulgarizations in the popular press, it can seem even weirder than usual. Last week was a classic example: the computer that could figure out the answer without actually doing a calculation! (See Uncertain Principles, Crooked Timber, 3 Quarks Daily.) The articles refer to an experiment performed by Onur Hosten and collaborators in Paul Kwiat's group at Urbana-Champaign, involving an ingenious series of quantum-mechanical miracles. On the surface, these results seem nearly impossible to make sense of. (Indeed, Brad DeLong has nearly given up hope.) How can you get an answer without doing a calculation? Half of the problem is that imprecise language makes the experiment seem even more fantastical than it really is -- the other half is that it really is quite astonishing. Let me make a stab at explaining, perhaps not the entire exercise in quantum computation, but at least the most surprising part of the whole story -- how you can detect something without actually looking at it. The substance of everything that I will say is simply a translation of the nice explanation of quantum interrogation at Kwiat's page, with the exception that I will forgo the typically violent metaphors of blowing up bombs and killing cats in favor of a discussion of cute little puppies. So here is our problem: a large box lies before us, and we would like to know whether there is a sleeping puppy inside. Except that, sensitive souls that we are, it's really important that we don't wake up the puppy. Furthermore, due to circumstances too complicated to get into right now, we only have one technique at our disposal: the ability to pass an item of food into a small flap in the box. If the food is something uninteresting to puppies, like a salad, we will get no reaction -- the puppy will just keep slumbering peacefully, oblivious to the food. But if the food is something delicious (from the canine point of view), like a nice juicy steak, the aromas will awaken the puppy, which will begin to bark like mad. It would seem that we are stuck. If we stick a salad into the box, we don't learn anything, as from the outside we can't tell the difference between a sleeping puppy and no puppy at all. If we stick a steak into the box, we will definitely learn whether there is a puppy in there, but only because it will wake up and start barking if it's there, and that would break our over-sensitive hearts. Puppies need their sleep, after all. Fortunately, we are not only very considerate, we are also excellent experimental physicists with a keen grasp of quantum mechanics. Quantum mechanics, according to the conventional interpretations that are good enough for our purposes here, says three crucial and amazing things.
First, objects can exist in "superpositions" of the characteristics we can measure about them. For example, if we have an item of food, according to old-fashioned classical mechanics it could perhaps be "salad" or "steak." But according to quantum mechanics, the true state of the food could be a combination, known as a wavefunction, which takes the form (food) = a(salad) + b(steak), where a and b are some numerical coefficients. That is not to say (as you might get the impression) that we are not sure whether the food is salad or steak; rather, it really is a simultaneous superposition of both possibilities.
The second amazing thing is that we can never observe the food to be in such a superposition; whenever we (or sleeping puppies) observe the food, we always find that it appears to be either salad or steak. (Eigenstates of the food operator, for you experts.) The numerical coefficients a and b tell us the probability of measuring either alternative; the chance we will observe salad is a^2, while the chance we will observe steak is b^2. (Obviously, then, we must have a^2 + b^2 = 1, since the total probability must add up to one [at least, in a world in which the only kinds of food are salad and steak, which we are assuming for simplicity].)
Third and finally, the act of observing the food changes its state once and for all, to be purely whatever we have observed it to be. If we look and it's salad, the state of the food item is henceforth (food) = (salad), while if we saw that it was steak we would have (food) = (steak). That's the "collapse of the wavefunction."
You can read all that again, it's okay. It contains everything important you need to know about quantum mechanics; the rest is just some equations to make it look like science. Now let's put it to work to find some puppies without waking them up. Imagine we have our morsel of food, and that we are able to manipulate its wavefunction; that is, we can do various operations on the state described by (food) = a(salad) + b(steak). In particular, imagine that we can rotate that wavefunction, without actually observing it. In using this language, we are thinking of the state of the food as a vector in a two-dimensional space, whose axes are labeled (salad) and (steak). The components of the vector are just (a, b). And then "rotate" just means what it sounds like: rotate that vector in its two-dimensional space. A rotation by ninety degrees, for example, turns (salad) into (steak), and (steak) into -(salad); that minus sign is really there, but doesn't affect the probabilities, since they are given by the square of the coefficients. This operation of rotating the food vector without observing it is perfectly legitimate, since, if we didn't know the state beforehand, we still don't know it afterwards. So what happens? Start with some food in the (salad) state. Stick it into the box; whether there is a puppy inside or not, no barking ensues, as puppies wouldn't be interested in salad anyway. Now rotate the state by ninety degrees, converting it into the (steak) state. We stick it into the box again; the puppy, unfortunately, observes the steak (by smelling it, most likely) and starts barking. Okay, that didn't do us much good. But now imagine starting with the food in the (salad) state, and rotating it by 45 degrees instead of ninety degrees. We are then in an equal superposition, (food) = a(salad) + a(steak), with a given by one over the square root of two (about 0.71). If we were to observe it (which we won't), there would be a 50% chance (i.e., [one over the square root of two]^2) that we would see salad, and a 50% chance that we would see steak. Now stick it into the box -- what happens? If there is no puppy in there, nothing happens. If there is a puppy, we have a 50% chance that the puppy thinks it's salad and stays asleep, and a 50% chance that the puppy thinks it's steak and starts barking. Either way, the puppy has observed the food, and collapsed the wavefunction into either purely (salad) or purely (steak). So, if we don't hear any barking, either there's no puppy and the state is still in a 45-degree superposition, or there is a puppy in there and the food is in the pure (salad) state. Let's assume that we didn't hear any barking. Next, carefully, without observing the food ourselves, take it out of the box and rotate the state by another 45 degrees. If there were no puppy in the box, all that we've done is two consecutive rotations by 45 degrees, which is simply a single rotation by 90 degrees; we've turned a pure (salad) state into a pure (steak) state. But if there is a puppy in there, and we didn't hear it bark, the state that emerged from the box was not a superposition, but a pure (salad) state. Our rotation therefore turns it back into the state (food) = 0.71(salad) + 0.71(steak). And now we observe it ourselves. If there were no puppy in the box, after all that manipulation we have a pure (steak) state, and we observe the food to be steak with probability one. But if there is a puppy inside, even in the case that we didn't hear it bark, our final observation has a (0.71)^2 = 0.5 chance of finding that the food is salad! So, if we happen to go through all that work and measure the food to be salad at the end of our procedure, we can be sure there is a puppy inside the box, even though we didn't disturb it! The existence of the puppy affected the state, even though we didn't (in this branch of the wavefunction, where the puppy didn't start barking) actually interact with the puppy at all. That's "non-destructive quantum measurement," and it's the truly amazing part of this whole story. But it gets better. Note that, if there were a puppy in the box in the above story, there was a 50% chance that it would start barking, despite our wishes not to disturb it. Is there any way to detect the puppy, without worrying that we might wake it up? You know there is. Start with the food again in the (salad) state. Now rotate it by just one degree, rather than by 45 degrees. That leaves the food in a state (food) = 0.999(salad) + 0.017(steak). [Because cos(1 degree) = 0.999 and sin(1 degree) = 0.017, if you must know.] Stick the food into the box. The chance that the puppy smells steak and starts barking is 0.017^2 = 0.0003, a tiny number indeed. Now pull the food out, and rotate the state by another 1 degree without observing it. Stick back into the box, and repeat 90 times. If there is no puppy in there, we've just done a rotation by 90 degrees, and the food ends up in the purely (steak) state. If there is a puppy in there, we must accept that there is some chance of waking it up -- but it's only 90*0.0003, which is less than three percent! Meanwhile, if there is a puppy in there and it doesn't bark, when we observe the final state there is a better than 97% chance that we will measure it to be (salad) -- a sure sign there is a puppy inside! Thus, we have about a 95% chance of knowing for sure that there is a puppy in there, without waking it up. It's obvious enough that this procedure can, in principle, be improved as much as we like, by rotating the state by arbitrarily tiny intervals and sticking the food into the box a correspondingly large number of times. This is the "quantum Zeno effect," named after a Greek philosopher who had little idea the trouble he was causing. So, through the miracle of quantum mechanics, we can detect whether there is a puppy in the box, even though we never disturb its state. Of course there is always some probability that we do wake it up, but by being careful we can make that probability as small as we like. We've taken profound advantage of the most mysterious features of quantum mechanics -- superposition and collapse of the wavefunction. In a real sense, quantum mechanics allows us to arrange a system in which the existence of some feature -- in our case, the puppy in the box -- affects the evolution of the wavefunction, even if we don't directly access (or disturb) that feature. Now we simply replace "there is a puppy in the box" with "the result of the desired calculation is x." In other words, we arrange an experiment so that the final quantum state will look a certain way if the calculation has a certain answer, even if we don't technically "do" the calculation. That's all there is to it, really -- if I may blithely pass over the heroic efforts of some extremely talented experimenters. Quantum mechanics is the coolest thing ever invented, ever. Update: Be sure not to miss Paul Kwiat's clarification of some of these issues.