Whenever  someone asks how quantum computers work, the first thing I do is point out that your regular computers are already working due to quantum effects. But of course, an answer like that is more than just tongue-in-cheek. Quantum computation, unlike what we are used to, uses quantum bits, or qubits, to perform calculations. This requires some explanation…

Because this is a "would" question, we can make some speculations. Right now, the best guess is that we would prepare some collection of quantum particles in some predetermined state. Then, we would use the electromagnetic field to manipulate these collections, and measure the final state to get the answers we want. This scheme has some nice properties. Since the advent of quantum mechanics, we can now control the production of electromagnetic fields to a level that Tesla could only dream about, and we can ignore the quantum corrections of light if we do it on a collection of quantum particles.

I describe this scheme because it makes it easier for you to see what is happening. Really, we would be expecting to manipulate the quantum particles' angular momenta, with emphasis on the intrinsic spin. The essential features are independent of the scheme used, because all of these systems would be governed by the same quantum mechanical equations.

Also an electron's spin only purely gets you -1/2 or +1/2 hbar values, a collection of them can average out to any intermediate value. This property makes it seem like we are abandoning binary computation for analogue computation.

In many ways, this is quite true. But then again, analog computers were invented before binary computers took root and, in museums, you can find computers that use the electric field (most notably, cases similar to the oscilloscopes still used in laboratories today). The earlier computer games were built like that too! But of course, they did not last. Binary computers aren’t as efficient when calculating physical problems, but we can do a lot more computation tasks, and it is so much easier to reset the machines for other computation. This is unlikely to be changed by quantum computation, so we do not expect massive changes to the paradigm of computation.

But the quantum computers will inherit the property of calculating physical problems better because quantum computing makes some calculations go faster. That is all—you don't get to calculate more things. But being faster is important, because the difference can be between actually being able to compute something (practically) or not. Otherwise, the strictly new part of quantum computation requires the reader to understand how quantum mechanics works. Contemplation of the consequences of Bloch Sphere of pure electron spin states is already so mind blowing that no short introduction can ever hope to do it any good.

So, I am sorry, I cannot do that part of the program justice.

Luckily, researchers in the field are very eager to tell you what they are doing. Even if they may get the physical pictures wrong, the results are just as useful. For example, the interested reader should check out Scott Aaronson's online lecture notes on quantum computing.

Sources and further reading:

• For really going into quantum computing, it is best to first get a good picture of quantum mechanics. Feynman Lectures Vol 3 is a thorough and clear introduction.
• Scott Aaronson's lectures are available here. They are, unfortunately, pompous in proclaiming the understanding of quantum mechanics, as pointed out earlier.
• I am currently under recommendation of my professor to study a book by Domenico D'Alessandro, "Introduction to Quantum Control and Dynamics". You must first be able to control a quantum system before computing can be discussed.
• It is, however, better to keep an eye on the present reality, and study how we manipulate quantum mechanical effects of semiconductors to calculate, without getting destroyed by decoherence. For this, one needs an introduction to solid-state physics, or the new name, condensed matter physics. To understand that introduction, however, one will have to go back and study statistical thermodynamics.
• And of course, if we are talking about normal computers, the theory of how it works now. Scott Aaronson does a brief review, because he needed to compare the computation power of quantum computer above what we already have.