As an prospective student, I am facing a crossing as many people did. I would love to know what is the current big question or motivation in the quantum information society? If anyone knows the answer for condensed matter or bio-physics or computational chemistry, you are welcome. Thanks!
Expanding upon what others have already said and adding some things, the main experimental challenges are quantum control:
Quantum communications: For cryptography, we need entanglement generation and maintenance ovver very large distances. Currently, there seems to be a barrier at about 400km due to decoherence and the quantum repeaters availabe. This needs to be overcome. 1st challenge: Build a good quantum repeater and demonstrate a quantum network over hundreds of kilometres. 2nd challenge: Implement satellite communications (might be easier than land-based communication for very large distances). Condensed matter people might be interested in repeater designs and implementations. If the challenges are overcome, one can begin to implement cryptographic schemes and think about industrialisation.
Quantum computers: Actually build a working quantum computer that can do anything new. This is the field where condensed matter physicists have a huge playground: The main problem for quantum computation at present relies in 1st challenge: creating good (high fidelity) multi-qubit gates and 2nd challenge: scaling the computer. No one needs a computer with 8 qubits. Condensed matter approaches might be best suited for the 2nd challenge once the first has been addressed, hence they are a very active research area. Both essentially means battling against decoherence. All research here ist still pretty basic, a working computer and what it might be useful for is at least 10 years in the future. Both challenges are essentially about quantum control.
Quantum simulatons: This is what quantum computing will actually be very good at. Simulating a chemical system is hard - especially because it is quantum mechanical. Why not simulate it using quantum mechanics? This idea is currently being implemented on certain smaller systems and might already produce results within ten years. Once again, the main challenges are achieving better control. Note that this is essentially reclaiming the domain of computational chemistry to physics. You'll need more physics than comutational chemistry to do this. Biophysics is still years away.
I see two issues regarding the subject.
The first one is technical implementations of quantum computers. I pretty much agree with Dirk on that. It is not easy to find a quantum system that is easy to control and is well isolated from the environment at the same time. There is still no agreement which quantum system is the best for quantum computing. It can be spin qubits, charge qubits or qubits based on photons. There are many attractive proposals. My favorite one is the Kane's spin qubits in silicon.
The second issue is related to what quantum computers can do in principle. Imagine that we have built one. What kind of algorithms can be efficiently solved on that computer? The problem is that, as far as I know, there is no an idea of the universal quantum computer, like the Turing machine in classical computing. There is a number of algorithms that might be efficiently solved on the quantum computers, but that is a pretty limited number. People are trying to find new such algorithms.