Secondly, how do quantum computers work?
Well, quantum computing is largely theoretical today. Although efforts have been made to make quantum computer and have been successful, a quantum computer of the sort that can completely outdo classical computer and be available in stores is not even on the drawing board!
A quantum computer is basically a computer that uses qubits instead of classical bits. A bit can have two spin states, 1 or 0. A qubit however, can have both of them simultaneously. This allows a few number of particles to carry a very large amount of information and manipulation of these with a laser beam can lead to calculations several thousand times faster than those of a classical computer.
On board a moon settlement, such a computer can be used to guide multiple docking operations that require multiple operations of similar type.
There are two major limitations of quantum computers:
a. They are difficult to built and maintain. The hardware has to be operated in extreme conditions, but with diamond spintronics (research this if you have time), it may be possible to make practical machines that operate in room temperature. Diamonds can yield multi-spin particles. Also, they have property called 'decoherence' i.e. these computers interact with their environments. This leads to errors.
b. Contrary to popular myths, a quantum computer has several limitations when it comes to computation. Computer scientists normally divide problems into different classes. Currently, algorithms exist for the P class. Class P problems are those which become more complicated by a certain power of a number as the number increases. This is the number involved in the computation. Quantum computers can outdo classical computers by performing type P calculations with less number of elementary steps, however there is no quantum algorithm for the hardest -- class NP-complete -- problems.
You can employ quantum computers on your settlement, predicting that the problems of decoherence will be resolved and diamond spintronics will enable practical quantum computers. At the same time however, you must identify the type P problems that you'll encounter in your colony. If you can identify such operations, you can justify the use and cost of quantum computers.