When we are talking about the quantum computation and classical computation, we are saying that quantum computation is exponential faster than the classical one. And that's because the Kronecker product of quantum states and quantum entanglement. Such as [1,0] state and [0,1] state will create a system state [0,1,0,0]. For a larger system, N quantum bits can store 2^N double precision numbers. And here the question raises, is quantum computation just an advanced data compression algorithm?
No, that is not how quantum computation works.
What you are referring to is a common misconception associated with quantum computation. Quantum mechanics cannot be used to store more data than what is classically possible. More precisely, it is not possible to use $N$ qubits to store more than $N$ bits of information in a useful way.
What is true is that, in general, to fully characterise the state of $N$ qubits you need to specify $2^N$ real numbers, and that simulating quantum systems is in general (expected to be) classically hard. This is however very different than saying that quantum mechanics allows for increased storage capability, which is wrong.