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While there are many interesting questions that a computer can solve with barely any data (such as factorization, which requires "only" a single integer), most real-world applications, such as machine learning or AI, will require large amounts of data.
Can quantum computers handle this massive stream of data, in theory or in practice? Is it a good idea to store the data in a "quantum memory", or is it better to store it in a "classical memory"?
It's not so much a matter of big data, but that of saving data. Quantum storage is still (much like the rest of the field) in its infancy.
(Take what I write with a grain of salt. It's likely to change rapidly.)
There are a few theories on how quantum computers might be able to hold "memory".
One of these is using nuclear spin. E.g. using long-lived nuclei in a quantum state. Converting an electron qubit (a qubit represented by an electron) to a nuclear qubit is possible.
A nucleus's coherence time - the time for which its phase is constant (when considering its wave function) - is longer than that of an electron. The linked article (same one as before) details how one can increase the coherence time of a nuclear spin (to some extent). The matter is being researched, but there is some indication that nuclear qubits can be a form of quantum storage.
The quantum state needs to remain, well, quantum. Additionally, if you entangle two of your "storage" qubits, you are likely to lose data.
Due to no-cloning, one cannot simply "copy" a qubit (whose state is unknown), which is one of the reasons quantum storage is difficult.
As for "big" data, it's just a matter of how much memory you have.
