Multi-qubit controlled gate with optimal T-count

Controlled gates are key components in various quantum algorithms. Improving on the prior work of Gosset et al., we show that, for an allowed error $\varepsilon$, $3\log_2\(1/\varepsilon\) + o\(\log(1/\varepsilon\))$ $T$ gates are sufficient to approximate most multi-qubit controlled SU(2)s. We also show that this T-count matches the lower bound when the use of an almost controlled gate is prohibited. As an application, general controlled gate synthesis and efficient SU(4) gate synthesis are given.