The number is far too large to have an SI name. Some people have put forward suggestions for new prefixes, but they wouldn't help you understand the number.

Edit: I suppose you could say 1 exayottayottabit, or 100 pebiyobiyobibytes, per cc.

And yes, this relies on using qubits for storage, in greater densities than is achievable.

There isn't a friendly name for 10^66 bits, but perhaps you can get an idea for the scale by considering the sizes of more commonly used names:

bit: 10^0, kilobit: 10^3, megabit: 10^6, gigabit: 10^9, terabit: 10^12, ... zettabit: 10^21, yottabit: 10^24

So we're at some pretty ridiculously large numbers and still nowhere near 10^66.