I replied to your message, olipoli, but I will try and write a more structured answer here.
Which is the most active field in terms of quantum information processing? In my opinion, it is physics, which is sensible because QC is largely a physics problem right now. EE is starting to edge in because we need people to build the devices, and there has been lots of work in EE to build nanoscale devices. Recently I was at the DAMOP conference where an EE from Sandia demonstrated that they have created a device that is capable of storing 300-400 ions (though of course not entangled). So the experimental physicists and EE's are working together to try and make this work.
In most respects, computer science isn't heavily involved in QC because QC just isn't quite there yet. Imagine that classical computing was around for a long time before we got around to abstraction away certain functions of a computer (remember, it goes from machine code to assembler level, to some other junk until you get to the typical languages, like Python or Java. In terms of algorithms, the algorithms are still very low level, and in the case of adiabatic quantum computation they need to be mapped to the specific topology of the device (think analogue computers). Still, algorithms are a huge aspects to QC. Recall that Peter Shor was the reason QC got so much attention, when he came up with Shor's Algorithm for factoring large numbers. The one thing that may help you as a CS person is that if you have some background in mathematical modeling, you would be valuable in the research sector where someone might want to simulate the behavior of a quantum computer for testing purposes. In fact, this is what I'm working on right now (I'm spending the next couple months at a national lab). I have a background in physics as well as programming and a focus on CS, so it is a good situation for someone with my skills and background.
So which of these will develop faster and have a better future? This is only my opinion, but I think the EE's will have it really good for a long time. If you're into building integrated circuits, you might find it interesting that D-Wave does superconducting ICs, and is actually hiring right now. Superconductors seem to be the best candidate for a practical QC right now, but D-Wave specifically is implementing an adiabatic QC, which is different than the gate-model computer that you may typically think of. Still, it uses qubits and may be able to solve difficult problems much more quickly than classical computers. But if you're working on quantum computing, you will likely know enough about the algorithms (if you're a good, competent researcher anyway) as an EE, as well as the ability to fabricate the actual devices, which makes you very valuable to any company who wants to buy, use, and maintain a QC (think Lockheed Martin, Amazon, McKinsey, anyone with logistics problems, etc.).
In response to your question about getting admitted to a CS program without having majored in it, yes you can. In fact, I am doing that very thing right now. I've been majoring in physics, and I'm planning on a Ph.D in CS (although probably not for quantum computing, though it's still possible... I'll have to wait to see how the field pans out for CS people). One thing you want to show the school is that you know the fundamentals of CS and you won't crash and burn when you take your first classes. Knowledge of basic programming languages (C, C++, or Java), data structures and algorithms (very important!), complexity/computability, architecture, and possibly networking (I've tried to put them in some sort of most to least important). Personally, I'll be studying for the CS GRE so that I'll have something to show to these schools that has to do with CS (some schools, namely GaTech, require a CS GRE, at least for PhD admissions). You could also take some classes in CS, the major ones anyway, and get good grades in them. Being an EE will also help you with admissions, being closer to CS than physics.
One last thing, you'll want to look at the link posted by chiro just above. I think it's an excellent resource. Because the field is small right now, you'll want to make sure you're applying to departments that are doing things that you are interested in. Are you wanting to work on the chips? The theory? The theory of the chips, or theory of computation itself? Algorithms? And are there enough people with your specific idea in mind? It would be nice to be rather open-minded about what you want to do, as that will give you more options. I say contact these professors also and see what they want from a potential grad student, and what sort of background will make you useful to them.
