Hmm, that's a strange question, IMO.
Anything you can currently do will be able to be done on a Quantum Computer. In other words, Quantum Computers will be able to emulate Classical Computers.
The question is strange because Quantum Computers are no where near the stage where people are thinking about what OSes are going to run on them. If you have a Computer Science background, relate it to Alan Turing having an idea about this Turing Machine he was going to make and how it could work, and then ask "Well, will it run UNIX?"
Quantum Computing will provide new methods of doing things that classical computers do, but most (if not all) of it is muddled up in theories and papers. UNIX may one day run on a Quantum Computer, but no one is thinking about that just yet. What the studies concentrate on is "How fast can a Quantum Computer Calculate the factors of this big number m compared to a Classical Computer?" or "How many queries in a given set of data will a Quantum Computer take to find an element n?", etc etc.
Hope that helps.