I don't think there are degrees in Quantum computing yet.
There are certainly courses but most of them tend to focus on the quantum mechanics of single bits (i.e. qubits), i.e. essentially "applied quantum mechanics".
Could you be more specific about what you are interested in?
Are you interested in "software" (error correction codes, gates etc) or "hardware" (the physics). The former seems to be done mainly by people with a background in math/CS and the latter by physicists specializing in whatever "platform" they are using (everything from liquid state NMR to superconductivity).
QC is already a huge, interdiciplinary, field.
If you want to get some idea about what the field is like I would recommend a book by Nielsen&Chuang (I don't remember the title) which is used by just about everyone in the field and in most courses on QC; it is also relatively easy to read and you do not need to know a lot about quantum mechanics to at least understand the basics.