Are qubits just analog, continuous classical bits?

Question

Topologically, classical bits (cbits) are essentially special cases of qubits restricted to the poles of the Bloch sphere. However, this restriction doesn't seem to be classical per se, but is simply inherited from the historical fact that transistors are either on or off. One could very well conceive of a classical bit which lives in a "classical" Bloch sphere of sorts, i.e., imagine some tiny, fully classical sphere with a needle that can point in all directions. Obviously, one of these dimensions won't be imaginary, but topologically speaking, it would be equivalent to the quantum Bloch sphere. E.g., if the needle were pointing along the equator, then we'd have a perfect example of classical superposition, which one could argue is "coherent" since it has a well defined phase and isn't the result of a statistical mixture.

I'll try to expand on the above with an example. Let's say our classical measurement basis is $NS$ (North $1$, South $0$) vs. $EW$ (East $1$, West $0$). To answer the question "is the wind blowing in the $NW$ direction?", we would need at least two measurements: One along the NS basis vector and the other along the $EW$ vector. If the results is $10$, then the answer is affirmative. However, would may as well apply a "classical Hadamard gate" which rotates the classical basis so as to answer the same question with a single measurement---as opposed to two---hence exhibiting the same "speedup" as what is typically purported to be unique to quantum superposition.

Ultimately, I'm looking for the "secret sauce" that qubits have that cbits don't. Clearly, there has to be more to it than what could be possible with my example of classical, analog Bloch spheres.

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Cross-posted on qc.SE

Answer 1

Are qubits just analog, continuous classical bits?

No. The continuous (vs. discrete) nature of qubit states is not the secret sauce that makes them so powerful. The secret sauce that is missing from a classical model is entanglement.

One minor failure of your proposed classical model of qubits is that (as far as I can tell) it doesn't have the update rule that measurements necessary force the system into the observed state. But that isn't the end of the world; you could add that in manually, and while it might not be terribly natural, that would reproduce that aspect of QM.

But the more important aspect that your model fails to capture is entanglement between qubits - more specifically, the exponential scaling of the Hilbert space dimension with the number of qubits. You're actually correct that a single qubit is essentially classical: it's a little-appreciated fact that the Kochen-Spekker theorem happens to fail for a single qubit, and there does exist a classical model that fully describes one qubit. But once you start combining multiple qubits together, the topology of the full system is not just the $n$-fold product of the 2-sphere, as it would be if qubits were fundamentally classical. Instead, it's a much higher-dimensional manifold $\mathbb{C}P^n$.

If qubits were classical spheres, then you could still describe correlations between them by simply defining a probability distribution over $(S^2)^{\times n}$. But such a classical probabilistic model wouldn't be capable of violating the Bell inequality, efficient quantum computing (as far as we know), and all of the other strange things that qubits can do. There just aren't enough degrees of freedom in your proposed classical model to make all of that stuff happen.

Answer 2

Your proposed qubit of an arrow on a sphere is a fair model of a single qubit, but it does not scale to multiple qubits. Your qubit's state can be described by a single complex number. If you had $N$ Of your classical spheres you could describe the total state with $N$ complex numbers, but $N$ qubits require $2^N$ complex numbers to describe the state. As explained in the other answer, this is because your classical model doesn't capture entanglement.

This is the exponential scaling of quantum resources. With your classical spheres you would be able to model $N$ qubits if you had $2^N$ spheres, but that's not really better than using a classical computer to model a quantum system, which doesn't scale that well.

It's true that the analog nature of quantum computers is part of the secret sauce that makes them powerful (it's also their achilles heel in that it makes them error-prone), but it's more than that. Another analog classical model that captures complex amplitudes would be discrete acoustic modes of a vibrating string, but again, what you miss out on is the exponential scaling due to entanglement. In quantum mechanics if you add one "thing" (a qubit) you double the dimension of the Hilbert space you model. As far as I can tell, if you try to make a classical analog linear algebra solver, you will need to add $2^{N-1}$ "things" to scale your model from $N-1$ dimensional to $N$ dimensional.