From what I understand, a qubit can store an infinite amount of information. So, where does he get the number 300 from? Why not just one? Also, can't the three-dimensional position of a particle can be classically recorded to the limit of infinite precision? Is there some limit at the Planck length? Any help/clarification would be appreciated.

A qubit is a superposition of two states: |0> and |1>, it's not an infinite amount of information, and you should really be precise with the terms you use.

Thanks for helping me be more precise with my terms! What is the amount of information stored in a superposition of |0> and |1>? Many sources claim this amount is infinite:

Are these sources incorrect? Or in need of more precise terms?

The major question I want to answer is, does Neil Turok have a valid point, or is he being deceptive? He's come up with the very specific number of 300 electrons. Knowing that a qubit is a superposition of two states, |0> and |1> doesn't help me understand how he got the number 300. Did he just pull that number out of thin air? Or does he have some practical quantum computer in mind?

Likely he's looking at 2^{300}, which is the number of states a collection of 300 up-or-down electrons can have. This is approximately equal to 10^{90} if you use the convenient rule of thumb that 2^{10} approximately equals 10^{3}.

The number 300 comes from a pretty good approximation.
In order to record the position of ##10^{90}## particles, you need ##10^{90}## registers, one for the position of each particle.
Now, ##10^{90} = 1000^{30} \approx 1024^{30} = 2^{300}##, that is, 300 bits of information. One qubit, as its name suggests, encodes one bit, and hence 300 qubits are required.

I'm going to call BS on Dr. Turok. He's much smarter than I am, so he is probably correct in some underlying way, but I think he is being misleading. A single qubit does not store more information than a normal bit. Assuming that a measurement basis has been chosen, the measurement will come back 0 or 1 just like a classical bit. If I want to communicate something with this bit alone, that is all I can do. I could put the bit in a superposition, but unless you know to measure a different basis, that won't help. So, in the sense of quantum computers, a bit stores the same information as a qubit. The power of quantum computing comes from the algebra, not more storage space.

Since a qubit has two free variables that can continuously vary (it is commonly pictured as the Bloch sphere which is just the surface of a sphere; a geometrically simple surface) one could argue that there are infinitely many possible states that a qubit could be in (like positions in the sky). That is true, but that a classical particle on a string has infinite information. That isn't really useful.

Also, the wiki article you linked to actually says that, from an information theory stand point, a qubit should not have infinite information.

Nugatory and Fightfish, thanks for your help reverse engineering this claim! This was just what I was looking for. I feel vindicated, thank you so much.

Yes, DrewD, I am happy to hear you came to the same conclusion as I did! Your explanation makes a lot of sense, thanks for adding some helpful ideas. I liked what you said about algebra vs. storage space. Also, a classical particle on a string. Good points!