Computability, universality and quantum automata

I will draw a distinction between finite-dimensional quantum evolutions ("automata") and infinite-dimensional evolutions ("operators"), and then explore their consequences upon two well-established concepts in Computer Science: computability and universality. Most of the results I will mention will rely on a decomposition of quantum operators, into quantum automata --- which is based upon the tacit assumption of a fixed partial order. Time allowing, I will try to touch on the topical question of quantum partial orders.