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I'll try to answer this by presenting two types of computations a quantum computer is good at.

1) Quantum Simulation. So in a quantum system, your state can be in a superposition of states ("because waves"). There is a lot of information that resides in the way this system is in superposition and it determines how the system will interfere and evolve (amplification or destruction of certain states). Mathematically, it means you need to keep track of the complex amplitude of each of the 2^n states of a n-qubits system. On a classical computer, this means 2^n complex numbers are necessary to represent the quantum state.

However, on a quantum computer, you only need n-qubits to represent the quantum state, because qubits can be in superposition the same way any quantum state in nature can be. You can now do careful operations on your n-qubits state and simulate any quantum mechanical phenomena. It is therefore way easier to simulate quantum mechanics on a quantum computer than on a classical computer. This could be useful in many areas, like chemistry to simulate protein-folding.

2) Factorization (Shor's algorithm). It is possible to engineer a quantum state that is the superposition of all possible inputs. Then, it is also possible to carefully create interference so that non-solution of the factorization problems get lower amplitude, while solutions get bigger amplitude. At some points you can measure the state and collapse it in one of the state. Since the amplitude of the solutions is big compared to the amplitude of non-solutions, you get the solution with significant probability. You can then verify it using any classical computer because multiplying two numbers is easy. This algorithm breaks most of today public key cryptography used on the internet.