In the paper Universal quantum computation with ideal Clifford gates and noisy ancillas, it is claimed that a circuit composed by Clifford gates, plus a so-called "magic state", can perform any quantum computation.
This paper is based on the formalism developed for fault tolerance. However, the claim above does not have to do with fault tolerance, nor with "noise". Indeed, the section III, which discusses the universal quantum computation, does not even mention the "noise" and the magic state is simply a well-defined pure state. It looks like it is possible to build any circuit using Clifford gates and simulating the T gate by using the so-called magic state, which is simply $\alpha\left|0\right>+\beta\left|1\right>$ with well defined $\alpha$ and $\beta$ (together with a particular procedure and additional Clifford gates).
So I would like to understand what I miss, if I do not understand the discussion about the fault tolerance and noisy qubits. Looking in literature, I feel that I actually miss something. For example, here I read this sentence:
quantum computers with magic states will most likely have the vast majority of its usable qubits be used for the distillation of magic states
Thus it seems that this "distillation" is needed: is it simply the preparation of $\alpha\left|0\right>+\beta\left|1\right>$ with known $\alpha$ and $\beta$?
So I would like to have a sketch of the idea of the relation between the discussion on universal computation and fault tolerance, or a reference discussing this idea more in depth but without relying too much on the language of fault tolerance.