So a really simple way of describing a digital computer is to say that it is a device for performing boolean operations. You feed it a bunch of bit strings, which is a description of the problem and its parameters in binary, the computer performs a bunch of boolean operations like $\wedge$, $\vee$, $\neg$ and gives you back another bit string which hopefully is the encoded version of the answer you were looking for. From this description it is not hard to see the connection of classical computation to discrete dynamical systems and classical logic, the operations provide the dynamics by flipping bits around in a controlled fashion and since we restrict the operations to a certain subset we get classical logic. My question is about analog and quantum computation. Is there a simple description of a quantum computer or an analog computer that makes the connection of that notion of computation to the other branches of mathematics a little more obvious? What is the most basic kind of enconding of a problem that can be fed into a quantum or analog computer and what are the most basic operations performed on this encoding?

Edit: Downvotes should come with comments so I know what to change in order to make my question clearer. Being a novice I'm just trying to piece together some themes about computation, logic and algebra.

Edit: alpheccar provided a link to a paper by John Baez and Mike Stay that pretty much answers my question and even puts it in a much wider context. Here's the link alpheccar provided.