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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"?

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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.

Why nuclear qubit/spin?

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.

What makes it difficult

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.

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