The thing to understand is that amplitudes are not actually just probabilities. For a single qubit, they are a pair of complex numbers a and b such that |a|^2+|b|^2=1. If you force a and b to be positive real numbers, then they really can be thought of as (square roots of) probabilities, and knowing one determines the other. However, since they can be complex, there are actually three real degrees of freedom, not one.
On the face of it, it’s not really clear what this buys you since measurement does force these complex numbers into their plain old, squared-norm probability. The interesting part is that there are quantum computational operations which only make sense in the complex world; there’s no way to realize them in the vanilla “probabilistic computing” world (that is, where all amplitudes are positive real).
Where things really start to get interesting is with “entanglement” of qubits, where now you have complex linear combinations of all possible n-bit states (where again the squared-norms must sum to 1). If you think of these linear combinations as elements of C^{2^n}, then quantum gates can be thought of as matrices acting on this space. Again, the “secret sauce” here is when you have gates that don’t come from classical computation (permutation matrices) or even probabilistic computation (real matrices with positive coefficients).
Two questions, for anyone who knows these things:
* Perhaps it's just not perfectly edited, but the above says the amplitudes are related to the probabilities that the qubit is 0 or 1. Shouldn't that say will be 0 or 1, because the qubit is currently in a superposition?
That may seem like nitpicking over typos, but it's necessary to the next question:
* Why have two amplitudes rather than one? The probabilities of either of two exclusive results, such as the two states of a resolved qubit, are P and 1-P. Given that amplitudes are directly proportional to those probabilities (in the last sentence of the above quote), how could the amplitudes change independently of each other? If the probability of one outcome increases, the other must proportionally decrease. And why not use one amplitude, because (it seems) if you know one amplitude you can easily determine the other? I suspect I'm misunderstanding something, perhaps that last sentence.