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The data is stored as charge in an extra component of a normal MOSFET transistor that is insulated all around. Roughly speaking, you have a sealed box inside of which is a conductor that can hold charge, and which is surrounded by insulator all around.
To "store" a bit in the box, you arrange for a higher than normal voltage to be applied across the insulating walls of this "box," and as a consequence electrons tunnel across into it. Once you remove the voltage, the probability of tunneling out is significantly reduced, and the bit stays "remembered" for a long time, without power being applied (which you need for RAM).
So the quantum aspect is getting the electrons in and out of the storage box, they tunnel in and out, which I would guess you can do with voltages low enough that they don't cause any damage to the box walls -- as, perhaps, an electron crashing through in a classical way would. Obviously you need very, very thin walls, but this is easily achieved in microlithography (chip-making).