A good collection of problems and their solutions, covering various topics in Quantum Information theory can be found in the following book:

Problems and Solutions in Quantum Computing and Quantum Information, by Willi Hans Steeb and Yorick Hardy

If you mean to ask what problems can be solved specifically by leveraging quantum computers, then cryptography, simulation and optimization will benefit a lot, which otherwise require a lot of raw computational power.

In my opinion it is deep learning of future neural networks systems as well.