Citation: Bravyi, Sergey, and Alexei Kitaev. "Universal quantum computation with ideal Clifford gates and noisy ancillas." Physical Review A 71.2 (2005): 022316.
Web: https://arxiv.org/abs/quant-ph/0403025
Tags: Foundational, Computer-scientific, Universal-scheme
This is the paper which introduces magic state distillation. A nice summary of what's left to do is given by the authors: "From the experimental point of view, an exciting open problem is to design a physical system in which reliable storage of quantum information and its processing by Clifford group operations is possible. Since our simulation scheme tolerates strong decoherence on the ancilla preparation stage, such a system would be a good candidate for a practical quantum computer."
For the surface codes, this is simple. You can prepare lots of (faulty) physical qubit T-states. Then, physical T-states are turned into logical T-states by state injection. None of this is fault-tolerant, so the result is a noisy ancilla. Magic state distillation can now give a not-noisy ancilla, completing the procedure!