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Presently, how much information can a quantum computer store, in how many qubits? What restrictions are there and how does it vary across realizations (efficiency of data storage, ease of reading and writing, etc)?

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Unfortunately the state of the technology regarding memories is not as developed as you seem to expect. When we talk about a memory, we think of a device that can store information for an infinite amount of time (for all practical purposes). So before we can think about the size of the memory in a quantum computer, we should look at whether a single quantum memory has been built. There is a lot of progress in this direction, but to my knowledge the currently best "memory" achieved a coherence time of about 6 hours (which is amazing, but still not what we are used from classical computers). Although the fidelity of the retrieved state is in the high nineties, the success probability for storage and readout is very low.

There is also work on using error correction codes to built a memory, but those approaches do not give better results so far.

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