Here's a (probably over-simplified) outline of quantum computing.
Quantum indeterminacy or "wierdness" means that a particle or small system can exist in two (or more) states at once, until you observe it. that "colllapses the wavefunction". Repeat the experiment for a two-state system, and it'll be like tossing a coin: heads half the time, tails the other half of the time. That's a quantum bit or qubit. Doesn't sound very useful, does it.
What a quantum computer does is to provide a quantum register (N qubits) and the means for performing mathematical operations on the contents of that register without observing it . This is equivalent to performing those operations on every possible N-bit number at once, from 0 to 2^N-1.
And then you observe it, and get just one out of the possible results.
However, consider a sequence of operations that will return one of a given number's prime factors. With 4 qubits, this has been done. Perform the sequence of operations that will return a prime factor of 15, then observe the quantum register, and you will get 5 or 3 with probability 1/2 each. All other possible 4-bit numbers have a probability of zero. They are not prime factors of 15 so they won't ever be observed.
With 32 quantum bits, it's looking rather interesting(*). With 4096 quantum bits, PKI is dead. For a billion quantum bits or a mole (~10^26) of quantum bits, you are performing magic(**). I do not believe that quantum computing can work for large numbers of qubits. Prediction: the upper limit of how many quantum bits one can work on, will tell us something very interesting about quantum mechanics, physics, and the structure of the universe. Of course, it's possible that the universe really is stranger than I can imagine and there is no upper limit....
Cryptography as we know it may or may not survive the experiment.
(*) AFAIK nobody has yet made even a 32-bit quantum computer. But if they had, would they be telling us?
(**)This is akin to finding a fast algorithm for solving NP-complete problems. (***)
(***) which, assuming you tell anyone else, is akin to signing your own death warrant. If you tell only a small number of people, some intelligence agency will take extreme measures to make it their secret. If you manage to spam it far enough and wide enough ... you probably just accelerate the rate at which a strongly Godlike AI bootstraps itself, and takes over the universe. Which given a universe containing billions of galaxies like ours, would almost certainly have happened already were it possible at all. So I predict that it isn't.