If quantum computing requires hundreds of digits of accuracy, how will it be possible? Leonid Levin said, "Exponential summations used in QC require hundreds if not millions of decimal places accuracy. I wonder who would expect any physical theory to make sense in this realm." See https://groups.google.com/forum/m/#!msg/sci.physics.research/GE5cz3xefCc/e0eh34MZGdwJ
Given that no machine has ever been designed to be sensitive to physical quantities to hundreds of digits of accuracy, how will quantum computing ever be possible in the real world?
To explain what I mean, in a QC model, the state vector has exponential size dimension in which the squares of the entries add to one. So if all of the entries are equal, when they are rounded to say the billionth digit, they will all be zero on a 100 qubit machine, contradicting the fact that they all must add to one. This is a big problem.
EDIT: I think the question is this: How can we possibly perform sensible measurements on quantum computers, given their extreme sensitivity?
 A: Craig, I think you're confusing two important things. First, your original question was something along these lines: 
Given the fact that we can expand a state ket $|\psi\rangle$ in term of basis kets
$$ |\psi\rangle = \sum_n c_n |\psi_n \rangle $$ 
and that there can be infinitely many $n$, let's consider a state which has equal probability to be in any of these (infinitely many) basis states. The question is, what happens when you measure this state? In QM, when you measure, you're supposed to get one of the basis states as the outcome for your measurement. 
Now here comes your point: You presume that, if the chance to be in any of the basis states is tiny enough, if you measure some sort of rounding error (?) will cause you to always not measure it in that basis state. You then reason that this will happen for each and every basis state, so that you can never measure $|\psi\rangle$ in any basis state, and we have a contradiction with the original statement that $|\psi\rangle$ could be expanded like we did.
Fortunately for quantum mechanics, your reasoning is flawed. If you measure a state, you will always get some outcome. There is no such thing as a rounding error in that sense in nature, this would contradict all sorts of continuity theorems and the likes.
Now, there is a more interesting, and relevant, question hidden in here. The fact is that quantum computers are extremely sensitive, and making controlled measurements without disturbing the system too much to ruin the computation is a legitimate problem in quantum computation. I added this to your question in an edit, for I think it's a question worthwhile posing. I hope this clears up the confusion for everyone involved.
A: If you believe the fault-tolerant threshold theorem for quantum computers, you do not require hundreds of digits of accuracy.  
Levin does not believe this theorem. More precisely, he believes that the hypotheses required for the theorem to work do not apply to the actual universe. 
I believe his mental model of quantum mechanics resembles the idea that the physics of the universe is being simulated on a classical machine which has floating point errors. I don't believe this is true.
A: Actually, it's possible, and all thanks to the use of more powerful processors, but not common processors, that kind of technology requires cold (or less heat) like about a few nanodegrees over 0 Kelvin. With the cold the particles are not so thick, so the electrons can flow easily trough the computer. Look for a video called NOVA MAKING STUFF COLDER
Also the size matters, if it's small is faster. LOOK TOO FOR NOVA MAKING STUFF SMALLER
