I'm not an expert in this subject area, but I know that I've sometimes seen the phenomena that are sometimes called "wavefunction collapse" described as non-unitary (and for good reason). Even if you buy into an interpretation of QM that doesn't include collapse as a separate process (most non-Copenhagen interpretations, in other words), measurements of the quantum state will still have to somehow look non-unitary to any given observer. (Something something projection operators, in the formulation I learned in grad school.)

There are definitely some interesting tricks you can do to exploit these behaviors. (I'm thinking for example of the "quantum Zeno effect", in which one can "find an answer without ever asking the question".[0]) But I don't know remotely enough about quantum computing to know what's actually possible in this context.

[0] A fun writeup on this idea is here (with an example involving an attempt not to wake any adorable sleeping puppies): http://www.preposterousuniverse.com/blog/2006/02/27/quantum-...