I am currently trying to understand magic state distillation. So far, my understanding is that the general idea is to find a code where a non-Clifford gate is transversal (very well explained in https://arxiv.org/pdf/1612.07330.pdf on Fig. 3).
For instance, that is why the 15-to-1 quantum Reed-Muller code is the most popular nowadays since as pointed out here https://errorcorrectionzoo.org/c/stab_15_1_3 it is the smallest stabilizer code with a transversal non-Clifford gate.
However, if you look at the paper which introduced distillation by Bravyi and Kitaev https://arxiv.org/pdf/quant-ph/0403025.pdf, they used the 5-qubit code. This code does not have a transversal non-Clifford gate. But I understand their protocol for why this reduces the error in their magic state.
So I finally come to my questions: Is magic state distillation with codes with a transversal non-Clifford gate just a particular case of distillation? If yes, what more general property than a transversal non-Clifford gate does a code require to be used for magic state distillation? Could we use any code to perform magic state distillation ? Or is there something I misunderstood?
Thank you !
