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I need to learn magic state distillation procedure and their application to fault-tolerance.
One of the original paper on this subject is the following: https://arxiv.org/pdf/quant-ph/0403025.pdf
I am wondering if there are more recent and pedagogic resource to learn this concept in an efficient manner (I know fault-tolerance with concatenated codes, but I need to learn how to do the non-clifford through magic states procedures).
Disclaimer: these links do not all fit the criterion recent.
First and foremost, it is important to realize that there is no 'one' magic state, and no 'one' magic state distillation.
The term 'magic state' is, loosely speaking, used to refer to any state that can be used, in conjunction with fault-tolerant Clifford operations and a specific quantum code, to create a fault-tolerant universal computer. One needs to add the (important) requirement that there exist an (efficient) distiller of these magic states so that they can be created themselves fault-tolerantly (using Bravyi's & Kitaev's method from the paper that you linked).
That being said, there are certain magic states that are more important than others. This presentation by Nathan Babcock has a nice introduction on the distillation process for 'T-type' states - I'll refer to the presentation for more details. There's a concise but nice list of references at the end of the talk.
The website of Earl Campbell has a nice didactic, higher-view introduction on 'magic'. There are some good references in there, too.
If you are a fan of (video) lectures, I can recommend the free-access(!) lecture series on quantum error correction by no other than Daniel Gottesman. For magic state distillation, check lecture 7(B). Edit: I just found out that there is an updated version from the lecture series in $2018$: here.
Finally, if you are interested in how magic states function within the scope of an entire fault-tolerant computer, check for instance this article by Campbell, Terhal and Vuillot.
