This is my first question on the Physics Stack Exchange; I hope it is not be a duplicate… :-|

I am not really a physicist (I am actually a professional mathematician), but I have already read quite a few things about quantum computation. And there is a problem that puzzles me: in quantum algorithms, it is assumed that you apply certain quantum gates like, say, the Hadamard gate:

$$H = \frac{1}{\sqrt{2}} \begin{pmatrix}1 & 1\\ 1 & -1\end{pmatrix}.$$

My point is: I do not doubt that you can apply some quantum transformations that are *very close* to the Hadamard gate, say

$$\tilde{H} = \frac{1}{1.414215} \begin{pmatrix}0.999999 & 1.000001\\ 1.000001 & -1.000002\end{pmatrix};$$

but getting a *perfectly precise* Hadamard gate seems as impossible to me as, say, building a corner reflector with *perfectly right* angles between the three mirrors (not that even *defining* what the angles are worth is meaningless below a certain precision level!). Moreover, as far as I understand it, certain quantum algorithms (including the Shor factorization) require so many successive applications of the Hadamard gate that even a $10^{-100}$ error in building such a gate would ruin the result of the algorithm…

The fact that surprises me most is that I have *never* read *anywhere* considerations on that issue of the precision of operators!—I have found *much* stuff devoted to the problem of interactions with the outside and decoherence, but nothing about the problem that I am concerned with…

Can anyone help me? Am I missing some fact which makes the precision problem actually irrelevant? Or, if that problem is relevant, are there some references about this issue? Maybe a classification of how much different algorithms would be sensitive to that matter…?