Its very easy to pull a logic gate or transistor circuit of the internet for example and begin to understand how it represents physical bits, by either having 5V across the circuit or below 3.3V. But I was wondering what are some of the methods for representing qubits from a hardware perspective.
I have already looked on Wikipedia and find the explanation to be very abstract and vague.
(This post has a bias toward superconducting qubits because that's what I know best. I encourage edits to add more information for the other qubit types)
There are many realized and proposed physical systems which can be qubits. Before we enumerate some of them, we should list the properties of a physical system required to actually be a quantum bit:
The system must have two quantum levels $|0\rangle$ and $|1\rangle$ separated by a frequency $\omega_{10} \equiv (E_1 - E_0)/\hbar$. Driving the system at $\omega_{10}$ induces transitions between the levels, which is how we do single-qubit logic gates.
The system must be at a temperature low enough to not scramble the quantumness. In particular, this means that we need $T \ll \hbar \omega_{10} / k_b$ where $k_b$ is Boltzmann's constant.
The system must not be coupled strongly to any environmental degrees of freedom. If the qubit couples to the environment too strongly, the qubit's quantum state decoheres. I've written a full example of how decoherence works and why it's hard to avoid in large systems here.
We must be able to control the qubit. For example, we have to be able to controllably induce transitions $|0\rangle \rightarrow |1\rangle$.
We have to able to controllably couple the qubits to one another so that we can do multi-qubit logic gates. This is not unlike a normal classical computer with transistors: in order to actually process information, we need to physically interact the voltages/currents of the transistors (i.e. in the CPU) to do AND, OR, and other operations.
We have to be able to measure the state of the qubits.
Now let's list some of the most successful qubit types.
This system uses the strong electromagnetic fields of an optical laser to trap ions in space. The traps keep the ions from interacting with the environment, and because The ions have orbiting electrons, and these electrons have various levels. We choose a pair of electron levels to act as the qubit states $|0\rangle$ and $|1\rangle$. The pair of states is selected to have a high coherence, i.e. a low rate of spontaneous emission. This is possible with atoms because they're so tiny: the electron transitions can have very small dipole moment and so they don't couple strongly to the electromagnetic field.
Logic gates are done with laser or RF pulses which cause the ion's electron state to change. Two-qubit gates often use the mechanical vibrational modes of the atoms in their traps.
Ion traps have been constructed using a variety of different atomic species.
The literature is vast. See here for a review of the prospects for trapped ions used for quantum simulation. See the groups of Chris Monroe, Reiner Blatt, and Dave Wineland.
An electrical $LC$ resonator has the Hamiltonian of a harmonic oscillator: $$H =\frac{1}{2} \frac{\Phi^2}{2L} + \frac{1}{2} \frac{Q^2}{2C} \qquad [\Phi, Q] = i \hbar \, .$$ This has equally spaced energy levels with spacing $\omega = 1/\sqrt{LC}$. It turns out that if you actually build an $LC$ resonator out of superconducting metal and cool it down so that $T \ll \hbar \omega / k_b$, it actually behaves as a quantum harmonic oscillator! You can then use it as a qubit... almost. Equally spaced energy levels are not good because you can't controllably use just two levels. To fix this, we add a nonlinear element called a Josephson junction to the circuit; this causes the levels to be unequally spaced, and we then use the bottom two as the qubit.
There are several different flavors of superconducting qubit, each designed for differing trade-offs between important figures of merit including: coherence, control accuracy, and reproducible fabrication. Some common flavors are the transmon, flux qubit, and fluxonium.
Superconducting qubits are pretty big, i.e. hundreds of microns, so decoupling them from their environment is harder than with trapped ions. Still, superconducting qubits on 2D chips routinely have energy lifetimes ($T_1$) of 40 microseconds, and in my group (Google/UCSB) two-qubit entangling gate times on the order of 70 nanoseconds. This means the qubits can do ~570 gates in one energy decay lifetime. Gate fidelities are just on the cusp of being good enough for error corrected quantum computation. Readout accuracy of around 99% has been done in less than 200 nanoseconds.
The links and details in the previous paragraph are from my own group, because that's what I know best, but there are many groups using different approaches to superconducting qubits. See papers by the Schoelkopf lab at Yale, the Martinis lab at Google/UCSB, the DiCarlo lab at Delft, the Devoret lab at Yale, and DWave.
Electrons are natural qubits: they have a spin degree of freedom, which is automatically a quantum two-level system. They're small, so in principle they should have good coherence. The trouble is getting them to sit still and controlling them. To do this, folks use electrons embedded in solid crystals, often silicon. This has the downside that the electrons are therefore really close to a bunch of other atoms, so coherence becomes an issue.
Anyway, many groups are pursuing this route, including the Yacobi group at Harvard, HRL (they have papers, but I can't find a website for the spin qubit group itself), Charlie Marcus (not sure where his group page is), and probably a lot more I don't know about.
Actually the point of computing in general is that the symbols are abstract and intrinsically meaningless and devoid of physics; this lets us fill in any physical realization of those symbols that we want: as long as we can connect "this symbol over here is represented by this set of those other symbols over there" appropriately, we can realize these symbols (which may be 0 or 1 but may be whatever we want as long as there's more than one of 'em) in any medium. So in your computer, bits are sometimes stored as voltages, sometimes as whether current will pass through a junction or not, sometimes as magnetic spins pointed in various directions on spinning metal platters, sometimes as notches in a spiral groove on a disk that need to be read out with a laser, sometimes as a partial regulation of how much light is coming out of a single color from a single pixel on your LCD (or if you're reading this with e-paper, whether the black or white side of a little ball that's painted half-black, half-white is facing the outside of the display). This fundamental ambiguity about what the "symbols" really map to in the world allows us to use anything as long as we can connect the symbols over here to the symbols over there by some concrete rule.
That is why you're seeing such ambiguous answers in the context of quantum computing; it's because we intentionally defined "quantum" computing in a similarly abstract way and then we are allowed to implement it however.
Now just like you have "voltage" bits that you understand very well, let's take one particular quantum system to learn what qubits work like. And that is the spin-$\frac12$ system.
A particle with spin-$\frac12$, like an electron, has some intrinsic angular momentum which is deeply, unfailingly, quantum. On the one hand it is always "on average" spinning in a certain direction; on the other hand it is never "purely" spinning in any direction. This is because the spin measured along any given axis must take the form $\pm \frac 12 \hbar$ but the total angular momentum squared must take the form $L^2 = \hbar^2 \ell (\ell + 1)$ where $\ell=\frac12$ is what makes it a spin-$\frac12$ particle in the first place. So in some sense when we say that the electron is spinning in one particular direction, "up" for instance, we're really talking about the case where we've measured its spin along that axis as $+\hbar/2$, but we know that since it has all of this extra total-angular-momentum it has a distribution of spin in the "left" and "right" directions and "back" and "forward" directions that have been forced, essentially by the uncertainty principle, to be deeply uncertain, precisely because we're certain about the way that it's spinning.
In fact we know that the mathematics for the spin-$\frac12$ particle has these three matrices called the "Pauli matrices", $$\sigma_x = \begin{bmatrix}0&1\\1&0\end{bmatrix},~~~\sigma_y = \begin{bmatrix}0&-i\\i&0\end{bmatrix},~~~\sigma_z = \begin{bmatrix}1&0\\0&-1\end{bmatrix},$$ and these two complex numbers $[\psi_0, \psi_1],$ and you can find the average value of any measurement by finding the corresponding matrix $\hat M$ to that measurement and computing:$$\langle M \rangle = \begin{bmatrix} \psi_0^* & \psi_1^* \end{bmatrix} \hat M \begin{bmatrix} \psi_0 \\ \psi_1 \end{bmatrix}. $$ The identity matrix $I = \begin{bmatrix}1&0\\0&1\end{bmatrix}$ corresponds to measuring that the constant number 1 is still 1, which means that we must ensure that $|\psi_0|^2 + |\psi_1|^2 = 1.$ The Pauli matrices represent measuring the spin along the $x$, $y$ or $z$ axes, getting either $+1$ (spin angular momentum $+\hbar/2$) or $-1$ (spin angular momentum $-\hbar/2$) along that axis.
As you can see, the vector $|0\rangle = \begin{bmatrix}1\\0\end{bmatrix}$ where $\psi_0=1$ corresponds cleanly to a qubit-zero as well as to "spin-up" if the z-direction points upward. Similarly the vector $|1\rangle = \begin{bmatrix}0\\1\end{bmatrix}$ corresponds cleanly to a qubit-one as well as to "spin down". These are eigenvectors of the $\sigma_z$ operator, and they have no average spin in either the $x$ or $y$ directions, but again, remember that they do have components in those directions, as you can see by looking at $\sigma_x^2 = I$ for example.
In fact there is a famous "Bloch sphere" representation of a single qubit which represents the qubit in terms of three angles $\Phi, \theta, \varphi$ as: $$\psi_0 |0\rangle + \psi_1 |1\rangle = e^{i\Phi}\left( \cos\Big(\frac\theta2\Big) |0\rangle + \sin\Big(\frac\theta2\Big)e^{i\varphi} |1\rangle \right),$$ then notices that $\Phi$ has no effect on any expectation value above and can be discarded: the claim is that $(\theta, \varphi)$ are spherical coordinate angles locating an axis that the particle "most spins along." So that's the interpretation of the qubit as a point on a sphere, with $|0\rangle$ at the north pole and $|1\rangle$ at the south pole. (More on this: Wikipedia, A previous answer I wrote.)
Now the most simple single qubit operation would be to fire a charged spin-$\frac12$ particle through a magnetic field pointed in the $z$-direction. The Hamiltonian for this is going to look like $\epsilon ~ \sigma_z$ for some energy $\epsilon$ and the state always evolves like $\exp(-i \hat H t/\hbar)$ for a given Hamiltonian $\hat H$ acting on a time t, therefore defining $\alpha = \epsilon t / \hbar$ if this magnetic field applies to the particle for a time $t$ then it causes $$|0\rangle \mapsto e^{-i\alpha} |0\rangle, \\ |1\rangle \mapsto e^{+i\alpha} |1\rangle.$$Or, looking at the Bloch sphere we see that this maps $\varphi \mapsto \varphi + 2\alpha,$ with one of the $\alpha$ coming from the shift in $\Phi.$ You might prefer to simply do this dividing-out-$\Phi$ to the gates as well, in which case $|0\rangle \mapsto |0\rangle, |1\rangle \mapsto e^{i\alpha} |1\rangle$ is the "phase rotation gate" by $\alpha$, and again, it's just zapping the electron with a magnetic field so that its spin precesses about the $z$-axis.
We might similarly rotate it about the X or Y axes; in particular notice that $\sigma_x$ is basically the classical "NOT" gate mapping $|0\rangle \mapsto |1\rangle$ and $|1\rangle \mapsto |0\rangle.$
Now suppose that you have two such charged spin-1/2 particles, so two qubits, and it needs to be written as $\psi_{00} |00\rangle + \psi_{01} |01\rangle + \psi_{10} |10\rangle + \psi_{11} |11\rangle.$ You can either separate them and do these magnetic rotations to each in particular, e.g. doing the above rotation by $\alpha$ about $z$ to the second qubit effectively maps $|00\rangle \mapsto |00\rangle,$ $|01\rangle \mapsto e^{+i\alpha} |01\rangle,$ $|10\rangle \mapsto |10\rangle,$ and $|11\rangle \mapsto e^{+i\alpha} |11\rangle.$ Or else you can bring them near each other. If you bring them near each other then maybe they prefer to have opposite spins rather than parallel spins or vice versa, this gives you some ability to "entangle" their spin-states with a Hamiltonian $\epsilon \Big(|01\rangle\langle01| + |10\rangle\langle10|\Big),$ causing $$|00\rangle \mapsto |00\rangle \\ |01\rangle \mapsto e^{i\beta} |01\rangle \\ |10\rangle \mapsto e^{i\beta} |10\rangle \\ |11\rangle \mapsto |11\rangle $$ Now if we combine these, first off doing this operation, then rotating both qubits along the z axis with $\alpha = -\beta$, we find that $|00\rangle \mapsto |00\rangle$ and likewise for $|01\rangle$ and $|10\rangle$, but $|11\rangle \mapsto e^{-2i\beta} |11\rangle,$ giving us an implementation of the "controlled phase rotation" gate which is a little more well-known. It turns out that you now can express an arbitrary quantum computation in terms of these flying spins: a composition of gently bringing them together pairwise to entangle/disentangle, and rotating them by precession about various arbitrary axes, can be made to simulate any quantum-computing algorithm.
As quantum effects are "everywhere", there are many implementations. The problem is not how to build a quantum computer, but rather how good you can build one. One simple yet difficult problem is that a good implementation has be to interact with you (for you to read information) but not the environment (for it not to decohere easily).
Various examples are given in the textbook [1], chapter 7. Here is the snapshot of its TOC. Highly recommend you to check its content.
7.3 Harmonic oscillator quantum computer
7.3.1 Physical apparatus
7.3.2 The Hamiltonian
7.3.3 Quantum computation
7.3.4 Drawbacks
7.4 Optical photon quantum computer
7.4.1 Physical apparatus
7.4.2 Quantum computation
7.4.3 Drawbacks
7.5 Optical cavity quantum electrodynamics
7.5.1 Physical apparatus
7.5.2 The Hamiltonian
7.5.3 Single-photon single-atom absorption and
refraction
7.5.4 Quantum computation
7.6 Ion traps
7.6.1 Physical apparatus
7.6.2 The Hamiltonian
7.6.3 Quantum computation
7.6.4 Experiment
7.7 Nuclear magnetic resonance
7.7.1 Physical apparatus
7.7.2 The Hamiltonian
7.7.3 Quantum computation
7.7.4 Experiment
7.8 Other implementation schemes
