How is valid qbit information retrieved when measurement spoils it? As layman I couldn't yet conceive how one could know that a given qbit, if it is said to have certain probability amplitudes alpha and beta, actually have (fairly accurately) these amplitude values, since any measurements would have spoiled the qbit without furnishing the required informations, if I understand correctly. Could someone please help?
 A: Let's say you have a spin one half particle with a qubit ad the spin. It is in the state $\alpha\left| Z_{-1/2}\right\rangle+\beta\left|Z_{+1/2}\right\rangle$ and you'd like to verify that.
If you had millions of identical copies you could measure the z component and make sure you get $-\hbar/2,$ $\frac{|\alpha|^2}{|\alpha|^2+|\beta|^2}$ of the time and you get $+\hbar/2,$ $\frac{|\beta|^2}{|\alpha|^2+|\beta|^2}$ of the time. But this would not tell you the phases.
Instead you could note that. $\alpha\left| Z_{-1/2}\right\rangle+\beta\left|Z_{+1/2}\right\rangle$ is an eigenvector of $n_x\hat\sigma_x+n_y\hat\sigma_y+n_z\hat\sigma_z$ for some real unit vector $(n_x,n_y,n_z)$ with eigenvalue $+1/2.$ Then you can orient your Stern-Gerlach device in the $(n_x,n_y,n_z)$ direction and measure the spin. If you get anything other than $+\hbar/2$ then it wasn't in the state  $\alpha\left| Z_{-1/2}\right\rangle+\beta\left|Z_{+1/2}\right\rangle.$ However it now is in that state, even if it wasn't in it before.
And that's the real issue. If something was in the state $\alpha\left| Z_{-1/2}\right\rangle+\beta\left|Z_{+1/2}\right\rangle$ or in a state orthogonal to  $\alpha\left| Z_{-1/2}\right\rangle+\beta\left|Z_{+1/2}\right\rangle$ then you can orient your Stern-Gerlach device in the $(n_x,n_y,n_z)$ direction and measure the spin and find out which it is. But as long as the state isn't completely orthogonal to $\alpha\left| Z_{-1/2}\right\rangle+\beta\left|Z_{+1/2}\right\rangle$ then the $(n_x,n_y,n_z)$ measurement has a nonzero chance to put it into that state and of course thereafter it will act just like that state becsuse it now is in that state.
So you can't know for sure if it was in that state to begin with. But no one ever claimed you could. And this method gives the best chance (out of strong measurements) to find out if it was because its basically measuring how much is in that state, so it says yes 100% of the time when it was, says yes 0% of the time when it was orthogonal and gives an yes in an amount proportional to how much it is in that state compared to orthogonal to that state.
This is basically like measure the z component to find out if it is in the state $0\left| Z_{-1/2}\right\rangle+1\left|Z_{+1/2}\right\rangle$ and just like that can't tell if you were in $0\left| Z_{-1/2}\right\rangle+1\left|Z_{+1/2}\right\rangle$ or in $0\left| Z_{-1/2}\right\rangle+e^{i\theta}\left|Z_{+1/2}\right\rangle$ similarly we can't tell the overall phase only the relative phase of $\alpha$ and $\beta.$
And there is no way around that limitation.
A: I think Timaeus has covered the ontological question of what you can measure about a qubit, so let me expound a little more on the technological side. As Timaeus notes, if I give you a mystery qubit, you can't do a measurement that determines $\alpha$ and $\beta$. If I give you a million identical mystery qubits, you could measure them to get a pretty good sense of the absolute ratio $\left| \alpha / \beta \right|$, but to get the phase will require you to do more than just measure in the computational basis. For instance, you can tell the difference between $| 0 \rangle + | 1 \rangle$ and $| 0 \rangle - |1 \rangle$ by measuring in the $x$-basis.
As a practical matter, however, we try to set up our computations such that this doesn't matter. In general, if you want to talk about gate complexity or the like, you talk about a system that starts with all qubits in the $| 0 \rangle$ state and ends with a measurement in the computational basis. This standard makes sure that you can't "hide" the hardness of the problem in either initial state preparation or in final state measurement. So the only time these phases and amplitudes matter should be "inside" the quantum computer, between gates. Your question is then essentially, "how do we know the gates work?" (Note: I could say the same thing classically, sort of, since I don't make my computer report the bits of a computation throughout. I just kind of trust that the Mathematica plot that pops out at the end has been properly calculated. Measurement is the tricky bit.)
And the answer is that this is an open question, since we're still working on what sort of quantum gates we want to use in general. When people present results about their quantum gates, they generally do so by measuring its effect on a single qubit (be it superconducting, trapped ion, whatever) and measuring that state many times (Ramsey interferometry is common for determining phases). This then allows them to tell how close the unitary they're implementing is to the intended unitary (intended quantum gate). This is an important figure of merit for prospective quantum computing platforms and called the fidelity. The idea/hope is that we can construct scalable quantum systems in such a way that knowing it works when we isolate out each gate like this, we will also be able to say it works as well when we do many gates in a row. 
An important field related to this is the idea of quantum error correction, which allows you to detect that something has happened without measuring the qubit directly. Here, the idea is usually that you put a qubit into the machine, and make it so that you can detect if noise has flipped the qubit in a way that you didn't intend. So you don't know what the final state is (yet) but you know your physical qubit has picked up an extra $X$ operation from environmental noise, and you can undo this before proceeding.
A: In your question I see a confusion of several related ideas. So, let us examine how these things work in practice by following some researcher around as she tries to do some real physical experiments. I will be kind and only use quantum gates which have no imaginary components to them.
Theory part
First, our researcher uses the equations of quantum mechanics to predict some result. Here is a simple example: she may have come up with the following quantum logic circuit:

where H is the Hadamard gate, X is the classical-logic NOT gate imported into quantum logic, and the funny dots are "controlled not" (CNot) gates which flip 1 with 0 and vice versa on the "white circle", but only if the "black dot" is 1: they do nothing if the "black dot" is 0. The big idea is that the final X(3) H(3) at the bottom right can be either present (experiment $Y$) or absent (experiment $N$).
If the final `X(3) H(3) is absent, the laws of quantum mechanics predict the state
$$|N\rangle = \sqrt{\frac18} \Big[~|000\rangle + |001\rangle - |010\rangle + |011\rangle + 
|100\rangle + |101\rangle + |110\rangle - |111\rangle~\Big],$$
while if it is present, the laws of quantum mechanics reduce this considerably to
$$|Y\rangle = \frac12 \Big[~|000\rangle + |011\rangle + |100\rangle - |111\rangle~\Big].$$
The experimenter is intrigued for three reasons: 


*

*The third qubit has some early interaction with the second qubit, but then basically does nothing until it is observed, with or without this final X-H. 

*This final X-H can be delayed indefinitely into the future, after both 1 and 2 have been measured; and 

*The last two qubits in $|Y\rangle$ are perfectly matching in all 4 of those states. If you measure these two qubits you always get the same result.


Probing deeper, she discovers that if you post-select on qubit 3 being 0 (discarding all experiments where you measure it as 1), the first two qubits go from an entangled state in N, $|00\rangle - |01\rangle + |10\rangle + |11\rangle,$ to a disentangled state in Y, $|00\rangle + |10\rangle.$ The effective quantum state matrix of the first qubit in each case is therefore$$\rho^1_N = \frac12~\begin{pmatrix}1&0\\0&1\end{pmatrix};\;\;\rho^1_Y = \frac12 \begin{pmatrix}1&1\\1&1\end{pmatrix}.$$
She thinks about the most easy way to show off these entanglements: a double-slit screen where qubit 1 causes either a pattern $|\psi_0(z)|^2$ or a pattern $|\psi_1(z)|^2$ on the screen labeled with coordinate $z$. A good double-slit experiment will then show the pattern $\frac12 |\psi_0(z)|^2 + \frac12 |\psi_1(z)|^2$ for the first state matrix N, but the pattern $\frac12 |\psi_0(z) + \psi_1(z)|^2$ for the second state matrix Y. The first looks like two overlapping bell curves; the second looks like a wavy interference pattern.
So the prefactors of $\frac12$ and $\sqrt{\frac18}$ have been exact because this is an abstract mathematical calculation assuming that all of these components are perfect.
Experiment design
Now as I hinted above, the above quantum-circuit is that it actually has a straightforward experimental interpretation, which is presumably why our physicist was studying it in the first place: 


*

*The first component Hadamard(2), CNot(2,3), X(3) puts the second two qubits in the state $\sqrt{\frac12}\big[~|01\rangle + |10\rangle~\big],$ which is pretty common in quantum optics: it is the entangled-polarization of two photons emitted from a certain class of nonlinear crystals, where two photons are created together, one with polarization up-down and the other with polarization left-right.

*The first middle component Hadamard(1) can be interpreted as sending one of these photons through a double-slit experiment, so that its "which-way" information is stored in qubit 1. So $|0\rangle$ reflects going through slit $0,$ and $1$ reflects going through slit $1.$ We will say that this photon is the one from qubit 2: its polarization is therefore stored in qubit 2, but the path it takes through the two slits is stored in qubit 1.

*The entangling operation CNot(1, 2), Hadamard(2) is a slit-dependent rotation of the polarization. In this case, slit 0 rotates the polarization by +45° and slit 1 rotates the polarization by -45°. So we can instantiate this with two "quarter-wave plates," one directly in front of each slit, with opposite orientations.

*The final X-H operation corresponds to another quarter-wave plate which we can either insert or remove in front of our final polarization measurement for qubit 3. We moreover send this photon through a very long length of fiber-optic cable before it is detected, so that our computer has completely recorded qubit 1 before we even decide whether we're going to put the quarter-wave plate into qubit 3.

*We can measure qubit 3 with a polaroid filter and a photomultiplier tube; we can measure qubit 1 with the $z$-screen that we talked about before. 


We can screen the results of 1 with 3 with a "coincidence counter" between the two to make the results more mysterious-looking: we get to choose at an arbitrary time and distance later whether what we have already measured is a wave-like "interference pattern" (quarter-wave-plate: yes) or is simply a particle-like "two overlapping bell curves" (quarter wave plate: no). Our choice on qubit 3 appears to travel back in time to affect qubit 1.
More importantly: If we remove the pair of quarter-wave plates performing CNot(1,2), Hadamard(2) operation in step 3 above, the resulting measurement of Hadamard(1) simply yields an interference pattern always. The "which-way measurement" (which is what it would be if we had un-entangled polarizations!) appears to have "destroyed quantum interference" but our choice of a quarter wave plate or not is a "delayed-choice quantum eraser" which erases this earlier destruction or not at an arbitrary later time.
Actual experiment
Okay, now that she has actually designed the experiment, she must actually build the device. And that is where there are several hurdles. For example, we need lots of measurements to build up the wavy interference pattern -- a given photon will only register on one of the detectors. Moreover, the detectors are going to be independently firing due to noise, so sometimes by the time we've chosen to select what we're measuring for qubit 3 and measure it, two detectors will have gone off. The obvious solution there is to shorten the fiber-optic cable, which also improves photon-losses in that side of it, but this effect cannot necessarily be totally discounted and will probably create noise (when the real particle hits between detectors and some noise-particle gets registered instead). Maybe the quarter-wave plates are not 100% identical. Maybe the photon has rotated somewhat inside of the fiber-optic cable and we just need to rotate the polaroid screen to the ad-hoc direction that shows us the most vivid interference pattern.
If she persists in trying to correct all of these things, she might be able to get the noise down as far as one team did:

(Image taken from this famous Delayed-Choice Quantum Eraser experiment.) Here we are looking at the right-hand side, where something which is supposed to be two flat graphs is not quite, and something which is supposed to look like two $\sin^2$ or $\cos^2$ graphs also isn't. But the graphs were demonstrative enough for publication, so it doesn't matter much.
Conclusion
Quantum mechanics is a scientific theory. This does not mean that it is some "guess," in fact it cannot even be decisively proven wrong, per se. Rather, it means that it exists at a more abstract level: it is a language we use. Within that language, we can phrase models which predict the world. One component of this language is complex numbers (infinite-precision) used as "amplitudes". 
Of course, we adjust models until they are correct, so: the key mark of a good theory is that the corrected-models that it creates are relatively simple. That's true in this case: the above quantum logic circuit is remarkably compact for something which violates our common-sense intuition so much. You can try to do that with Newtonian mechanics, but it will involve an insanely complicated model of reality which probably tries to ultimately reproduce quantum theory: James Clerk Maxwell, the great unifier of electromagnetism, saw space as filled with little invisible cogs and gears, causing the light-interference effects that quantum mechanics puts straightforwardly in these amplitudes of wavefunctions.
When the theory meets the model and the model collides with reality, of course you're going to suffer from limited precision and noise and all sorts of similar crap. But the theory doesn't have to incorporate that: you can either design better experiments with less noise or else model the sources of noise with the theory as well.
