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 $\epsilon$, $3{\log }_2(1\mathrm{/}\epsilon )+o(\log (1\mathrm{/}\epsilon ))$ $T$ 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.