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I'm interested in the model of quantum computation by magic state injection, that is where we have access to the Clifford gates, a cheap supply of ancilla qubits in the computational basis, and a few expensive-to-distill *magic states* (usually those that implement S, T gates). I've found that the best scaling is logarithmic in the accuracy $\varepsilon$, specifically $O(\log^{1.6}(1/\varepsilon)$ is what a 2012 paper offers to get the accuracy we need in the $S,T$ states.

Is this enough to calculate most of the problems we're interested in? Are there any problems that specifically resist QCSI (Quantum Computation by State Injection) because of high overhead, but are more solvable in other models of computation?