The word 'state' makes sense in Quantum Mechanics. In classical computers, numbers are represented in binary as a series of 0s and 1s. It seems that the word 'state' extracted from QM and the 0s and 1s extracted from Computing are put together to make $\lvert 0\rangle$ and $\lvert 1\rangle$. This doesn't make sense. What's the physical significance behind that representation?

Consider: in classical computing, what do the 0 and the 1 refer to?

Sometimes (and as suggested by Turing when he introduced Turing Machines), they represent literal symbols which are written down: a '0' character, or a '1' character. But when we use electronic computers, they more often refer to a low voltage versus a high voltage, or a magnetic field pointing in one direction or another. In each of these cases, what we are looking for is some physical system, in which there are some 'states' (either ink on a piece of paper, voltages in a wire, or direction of a magnetic field) which we can easily distinguish from one another (by seeing the differences in the ink patterns under ambient light, or by a suitable piece of electronics).

Here, as in quantum mechanics, we are considering questions of the state of a physical system, and the reason for this is that information must always be encoded in terms of physically distinguishable properties of the ways that a system could be — properties of the state of the system.

When we consider quantum computation, we have the same situation, only we have to be much more precise in what we are distinguishing. We not only want to have easily distinguished properties — like the orientation of an electron spin, or the polarisation of a photon — but for those properties also not to be closely coupled to any other properties of the system.

If we can succeed in this, and if that relatively isolated degree of freedom can have only two distinct values (which we could give any labels that we like, such as value A and value B, or indeed '0' and '1'), then we identify this as a qubit. By virtue of it not being strongly coupled to any other properties of the system, we can consider any possible configuration that this isolated degree of freedom can have, so that we may consider superpositions of '0' and '1', whether we can cause this degree of freedom to interact in a controlled way with other measurable degrees of freedom of the system, and so forth.

Often, but not always, this isolated degree of freedom (our 'qubit') is a property of one particular part of our system, such as an electron, a nuclear spin, a photon, a current through a small superconducting element, etc.; and the fact that it gives us this qubit is due to the fact that it is not too strongly coupled to other electrons, photons, etc. in the physical set-up being considered. Of course, arranging that this should be the case is a question of delicate engineering, but we can consider how to do so. Even so, it is not necessary that a qubit be as easily isolated as pointing to one particular physical system: once we get going with quantum error correction, particularly if we use planar surface codes, the story of "where is my qubit" may become a bit more complicated — but the qubits will still be there, and will arise out of a well-controlled degree of freedom of the system.

In short: just as '0' and '1' are shorthand for physically distinguishable values of some degree of freedom in some classical physical system, |0⟩ and |1⟩ are shorthand for physically distinguishable values of some degree of freedom in a quantum mechanical system.

The |0,1> is a mathematical representation which makes correspondance with classical computing easier. But qubits can be realised by different physical systems.

Take for instance two different polarization of a photon; the alignment of a nuclear spin in a uniform magnetic field; two states of an electron orbiting a single atom...

Sometimes instead of 0,1, you can hear spin up or spin down to refer to the computational basis state. So it is just a matter of convention.

$\left|0\right>$ and $\left|1\right>$ are shorthand for vectors in a pre-defined state space. Their physical meaning depends on the underlying technology.

For example, you could have an optical polarization system, such that $\left|0\right>$ means that the photon is vertically polarized, and $\left|1\right>$ means that the photon is horizontally polarized.

You could have a superconducting flux loop system, such that |0> means that the loop current is clockwise (+ magnetic flux), and |1> means that the loop current is counter-clockwise (- magnetic flux).

You could have differential optical system, such that $\left|0\right>$ means that the photon is in the left fiber, and $\left|1\right>$ means that the photon is in the right fiber.

You could have a neutral atom system, such that $\left|0\right>$ means that the unpaired electron spin is parallel to the nucleus spin, and $\left|1\right>$ means that the unpaired electron spin is opposed to the nucleus spin.

Given the large number of possible technologies which might be used, you can see that this list goes on for a while.

The main thing to know is that, in principle, you could build a Quantum Computer out of any of these technologies. The software description should be identical, as long as certain elementary operations, such as CNOT, are supported. A quantum algorithm which implements an Oracle should work the same way, no matter which implementation technology is used.

As other people point out, this is similar to the situation in classical logic. Ethernet, for example, could be implemented using copper wire, optical fiber, or radio. This is called "Layer 1". The definition of 0 and 1 depends on the underlying technology. What is important is that Ethernet (and TCP/IP running on top of Ethernet) does not care what the physical implementation of the bits is.

Building upon the previous answers, I will give a concrete example in terms of spin-$\frac12$ particles.

Assume such a particle, say an electron, is at rest inside a magnetic field of strength $B$ pointing in the $z$ direction. The hamiltonian of this small system is

$${\rm H}=-\gamma B\,{\rm S}_z=-\gamma B\,\frac\hbar2\begin{bmatrix}1&0\\0&-1\end{bmatrix}$$

where $\gamma$ is some constant called the gyromagnetic ratio and ${\rm S}_z$ is the observable corresponding to spin in the $z$ direction. The eigenstates of this hamiltonian are the states that we prefer to name spin up and spin down, $$|\!\uparrow\rangle=\begin{bmatrix}1\\0\end{bmatrix},\;\;\;\;|\!\downarrow\rangle=\begin{bmatrix}0\\1\end{bmatrix},$$ with eigen-energies $$E_\uparrow=-\gamma B\frac\hbar2,\;\;\;\;E_\downarrow=+\gamma B\frac\hbar2.$$

If we were to build a quantum computer using these spin-$\frac12$ particles as qubits, we could relabel the spin up state $|\!\uparrow\rangle$ as $|0\rangle$, and the spin down state $|\!\downarrow\rangle$ as $|1\rangle$. After all, it is much more convenient to reason about strings of qubits like $|001001010111\rangle$ than strings of ups and downs, i.e. $|\!\uparrow\uparrow\downarrow\uparrow\uparrow\downarrow\uparrow\downarrow\uparrow\downarrow\downarrow\downarrow\rangle$. Measuring a qubit would then mean measuring its spin, and finding either $E_\uparrow$ or $E_\downarrow$ to determine which state it collapsed to.

Of course, you could build a quantum computer with a completely different system under a different hamiltonian, but the same principle holds. Computer scientists talking about quantum computation generally toss out the entire idea of hamiltonians, energies and so forth because circuits carrying $|0\rangle$s and $|1\rangle$s are easier for them to understand, and because it abstracts away the underlying physical system.

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