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Reading about qubits, I see a lot of names related to waves from physics (amplitude, magnitude, phase, ...).

Does it mean that each qubit corresponds to some wave? If yes, what is the mapping between wave features and qubit features? For example, qubit has 2 numbers called as amplitudes. How can these be mapped to a wave? On the other hand, waves have some frequencies. Does it mean that a qubit has also some frequency?

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I don't believe that there is an immediate connection between waves and qubits.

Most of the terminology is probably due to the historical development of quantum mechanics. For instance, you may have heard about waves when talking about the two slit experiment, where you shoot electrons through a pair of slits and you get an interference pattern on the screen.

In that case, the Schrödinger equation tells you that the electron is described by a propagating wave with a specific frequency $f$ which depends on the electron's energy.

Then, you can define the wave function $\psi(x)$ with amplitude $|\psi(x)|$. The intensity of the wave at some position in space is $|\psi(x)|^2$, which is equal to the probability density of finding the electron at $x$.

$$\rho(x) = |\psi(x)|^2$$

A qubit state is defined as:

$$a|0\rangle + b|1\rangle$$

Here $a$ and $b$ are called amplitudes. This connects to the terminology developed for the double slit experiment in the sense that $|a|^2$ is the probability of finding the qubit in state $0$ and $|b|^2$ of finding the qubit in state 1. Hence the modulus square of the amplitude gives you the probability.

The qubit can represent something like the state of the electron on atom, where $|0\rangle$ corresponds to the electron being in the ground state (close to the nucleum) and $|1\rangle$ to the electron being in an excited state. However, qubits can represent other quantum systems as well, such as photon polarization!

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  • $\begingroup$ "Hence the modulus square of the amplitude gives you the probability." -> The probability of what? $\endgroup$ Sep 9, 2022 at 22:39
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  1. Wave-particle duality is a general feature of quantum mechanics, see e.g. the relevant Wikipedia page, and this post on physics.SE.

  2. A "qubit" is an abstraction of a two-dimensional quantum system. When you talk about qubits, you are not making any reference to the actual physical substrate that you are modeling. Any two-dimensional quantum system is "a qubit". The "amplitudes" of a qubit are the numbers we use to describe its state. I don't know what it would mean to "map these to a wave". You also shouldn't think of "waves" as in "a number of objects each one called wave", that doesn't really make much sense. When saying a qubit "has wave properties" we simply refer to the way it behaves in certain circumstances, not to there being some object called "wave" corresponding to it.

  3. The "wave properties" in quantum mechanics are reflected in the phenomenon of interference. There is not, however, some sort of "quantifier of waveness" of a state. Interference is just a foundational property of quantum systems. In some contexts wave properties might be more visible than in others, but any quantum state will display interference/wave phenomena under suitable circumstances.

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To some extent, you can think that many implementations of qubits can be interpreted in terms of waves, as any particle can be interpreted as such (known as wave-particle duality.)

I do not, however, believe you can generalize the mapping of qubit-related terms like "amplitude" or "frequency" to waves directly. For some implementations you will have frequency and amplitude (for ex. photonic qubits) but they do not have to mean the same thing in both contexts (quantum amplitude of a state stored in photons is not necessarily the same as the amplitude of any of the photons in that state etc.)

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