Can a CS grad student work on quantum field theory and string theory problems?
There are plenty of computer scientists working on quantum computation and quantum information so, yes, there should be plenty of opportunities to work on that sort of thing as a CS grad student.

I'm not aware of any computer scientists working on string theory but it's far from my area of expertise so it's possible it happens without my knowing about it. However, I've been to plenty of theoretical CS conferences where there were a couple of talks about quantum things but I don't remember any talks at all about string theory. And, of course, if you try to Google for something like "string theory computer science", Google thinks you mean strings of text.