An $N$-qubit system keeps $2^N$ pieces of information or should we say $2^{N+1}$?

Say my quantum register is of size $3$ qubits. This means I'll need $2^3$ complex numbers to describe all of its possible arrangements. But each complex number requires $2$ real numbers, so maybe I could say I'll need $2^{3}\times2 = 2^4$ real numbers. Is this correct?

Granted, a complex number is a number. But when we implement a complex number on any classical computer, we always reserve two numbers for it. A complex number is a number, but down to computer technology it really is a pair of numbers. When it comes to storing information, the matter is more about technology than abstract science.

I know in complexity theory it doesn't matter if $2^{N}$ or $2^{N+1}$, but I haven't seen any mention of this little detail anywhere and I've been reading every introduction on quantum computing I can find. Please do mention any source that you might know that has touched on this little technicality. Thank you!

• If you actually had a quantum register, you would not store its state as pairs of real and imaginary parts but as as the actual quantum state of the register… I'd say your question does not make a lot of sense. – Mariano Suárez-Álvarez Jan 12 '18 at 21:09
• It is an important sense neither of those: en.wikipedia.org/wiki/Holevo%27s_theorem – Qiaochu Yuan Jan 12 '18 at 21:14