But all of this seems in a tragic state at the moment. Allowing rampant misinformation and hype as to what these machines are actually capable of.
The quantum state of a system is not described by a simple real probability but by a complex number. This means that there is an additional degree of freedom that you can adjust in a quantum state that doesn't change its probability of occurring. The nature of entanglement means that there is constructive and destructive interference when you go through certain quantum operations.
Traditionally, a quantum word can be thought of as being a probability distribution over the various possible values of the word. You can build quantum gates that act like classical gates on the binary values, and it's this property that often causes people to describe a quantum computer as executing every possibility in parallel. The difficulty is that the result you get is sampled by the probability distribution, which means you need some sort of interference between the results to allow you to boost the probability of picking the answer you want. A quantum speedup is only possible when you can find the interference. In the case of Shor's algorithm (for factoring), the interference is basically a quantum variant on FFT; in the case of Grover's algorithm (finding x such that f(x) = y), the interference is something that redistributes the probabilities among a plane of the sphere.
Secondly, I think you're going too deep on this whole 'prediction' thing. The algorithms in quantum computing aren't trying to model something any more than a classical algorithm does. Currently, the notion of a quantum algorithm is just a series of special logic gates that act on qubits instead of bits. What the previous poster said above is correct. Quantum computers are like classical computers with randomization except instead of real probabilities it has what are called complex amplitudes. The way the complex amplitudes behave allow for the states to interact slightly different since they can now 'interfere' with each other to cancel each other out in very specific situations. This power gives quantum computers only slightly more power the classical ones (with randomization). (Most) experts don't think quantum computers can solve NP-hard problems. This power from the interference only helps in very specific problems.
This is done applying the Quantum Fourier Tranform on each 'a^x mod N' using x+1 complex roots of unity for that value of x. You can visualise this as x+1 arrows from the centre of a unit circle with an angle 2pi/(x+1). So, for x=3 you have 4 arrows pointing up, down, left, and right.
When you stack the arrows end to end, they will loop back to their starting point (3 will form a triangle, 4 a square, etc.). The key is that when multiplting this quantum fourier transform with the value of 'a^x mod N', the 1s (at the period) will occur at the same point around the unit circle only for the cases where the number of complex roots is a multiple of the period.
This has the effect that for the correct answer, the arrows are lined up together in a single direction, amplifying the number and thus the probability of selecting that number. For the others, the arrows cycle and loop back on each other, so stay fairly small, decreasing the probability they will be selected.
This gave me a real 'aha' moment - I've been trying to get an intuition for complex numbers for years, thanks.
PBS Infinite Series has several videos on quantum computing and Shor's algorithm: 1. https://www.youtube.com/watch?v=IrbJYsep45E "The Mathematics of Quantum Computers" 2. https://www.youtube.com/watch?v=12Q3Mrh03Gk "How to Break Cryptography" 3. https://www.youtube.com/watch?v=wUwZZaI5u0c "Hacking at Quantum Speed with Shor's Algorthm"
One source of confusion is that there are different types of proposed quantum computers. Some, like the D-wave machines, use stochastic measurement processes to arrive at probably-correct solutions. This is called "annealing", and it's a lot like what many machine learning algorithms do to try and find global minima in error function spaces, only the quantum mechanics gives it an edge in not getting stuck in local minima for various reasons.
These aren't universal computers - they only implement very specialized algorithms in a kind of abstract sense. It's sort of like how if you take a box of different sized rocks and shake it for a while, you'll eventually find that the small rocks have made their way to the bottom and the bigger rocks have drifted to the top (this is called granular convection). There's a sense in which this is like a search algorithm that helps you find the biggest rocks. The quantum version of this (very loosely speaking) would be if the rocks sorted themselves faster because the small rocks would sometimes just tunnel through the big rocks.
Using quantum effects to speed up annealing lets you do annealing faster than a classical simulation - but most problems aren't annealing problems. So present day quantum computers are specialized machines that do a kind of computation using quantum effects, but they can't run general purpose quantum algorithms. Think of them like co-processor chips that aren't quite CPUS.
That said, it's sort of hard to define what a general purpose quantum computer would even be (for various reasons), so maybe that's unfair. Anyway, D-wave machines can't run Shor's algorithm, which is one of the examples of the "exponential speedup" over classical computation.
The (claimed) algorithmic efficiency of quantum computing as it applies to quantum algorithms has to do with how much computation is done in each step.
We represent information in physical systems and process that information by making those physical systems interact in a way that affects a state space transformation. The amount of processing we can do in "one step" depends on the size of the state space of our data representation and the kinds of operations we can affect via physical interactions. Conventional computation involves a bunch of parallel systems that have 2 distinguishable states interacting pairwise through networks of logic gates that perform a limited number of transformations.
Systems of qubits have many more than 2^N distinguishable states (technically, an infinite number), and operations on them can be arbitrary linear transformations. So each quantum logic gate can do a lot more computation than a classical binary gate. Quantum algorithms basically translate information in the limited classical state space to the bigger quantum state space, do fancy operations, then project back into the classical space. It's sort of like data parallelism in the sense that if you can do a series of small operations or one big SIMD operation, the latter will generally be faster. Operations on qubits make use of physical parallelism that's only possible in the much bigger non-classical state space.