If you are talking about a PhD, then the idea of this is to work on one research project in which you have a specific focus for the next 3-7 years (depending on the field and nature of the project).
If you have to (or have the option of) take(ing) interdisciplinary or unrelated subjects, then that's another issue, but your focus will be on a project that becomes a lot more specific as a result of collaboration with you and your supervisor (primarily).
If the project is interdisciplinary and includes a specific focus with two disciplines then that's the nature of the focus, but otherwise you won't be able to do a double major PhD in the way that has an analog with an undergraduate degree.
For graduate school, you are accepted to a single program. While you can always take a class or two in another field, there's no such thing as a double major at the graduate level, and you cannot enroll in more than one program at a time. If you're interested in both computer science and physics, check out computational physics and computational science programs. Maryland has a good one.
This is likely a school specific issue. I know of at least 3 schools that allow you to pursue more than 1 curriculum in graduate school. I imagine there are more if you look.
Furthermore, there are schools that, in addition to allowing more than 1 curriculum, allow you to have more than 1 concentration. Of course, there are procedures and approvals required, and it's most likely to apply to MS rather than Ph.D for reasons stated by chiro and eri.
In the case of Ph.D, if your interdisciplinary study is not too obscure, chances are, the other department has faculty that is part of your own department's affiliated graduate faculty. That way, you remain in your department, but your advisor is in another department. Ph.D is when you narrow down your area and become the expert and is no longer where you divide your time trying to be the jack of all trades like undergrad.