How do Quantum Computers work? I have a decent knowledge of quantum mechanics, undergraduate level, but would like a description of how a quantum computer works and why it is important that superpositions of states, ie coherence, exists.
 A: The simple answer: It doesn't work at all, because there is no quantum computer at the moment. The only viable candidate D-Wave might be the first somewhat quantum computer, but is far from doing anything useful.
So you might want to ask "How would a quantum computer work" and the answer to that is that this question is way too broad. There are many different models of quantum computation which use quantum mechanics in very different ways - the models I can think of just now would be


*

*Circuit based QC (similar to reversible boolean circuits). 

*Measurement based QC (build up correlations in a lattice of e.g. spins and then do the computation by measuring the lattice sites).

*Quantum annealing and especially adiabatic quantum computation (the computation result is the ground state of a Hamiltonian and you change the Hamiltonian of a known system adiabatically, i.e. leaving the system in the ground state).

*Topological QC (using systems with anyons, i.e. two-dimensional systems, for computation. This results in robustness to errors).


Apart from that, there are a multitude of proposed implementations in many different systems, but I believe you are not so interested in that. In any case, have a look at Nielsen & Chuang to get an introduction to the physics and computer science part of quantum computation. 
There is actually some debate about what actually makes a quantum computation work other than that it is definitely due to being able to use entanglement and superpostions. To get an idea, you should have a look at an algorithm that outperforms classical algorithms such as Grover's search (see the book for an introduction).
