Catalytic $z$-rotations in constant $T$-depth

We show that the $T$-depth of any single-qubit $z$-rotation can be reduced to $3$ if a certain catalyst state is available. To achieve an $\epsilon$-approximation, it suffices to have a catalyst state of size polynomial in $\log (1\mathrm{/}\epsilon )$. This implies that ${\mathsf{Q}\mathsf{N}\mathsf{C}}_f^0\mathrm{/}\mathsf{q}\mathsf{p}\mathsf{o}\mathsf{l}\mathsf{y}$ admits a finite universal gate set consisting of Clifford+$T$. In particular, there are catalytic constant $T$-depth circuits that approximate multi-qubit Toffoli, adder, and quantum Fourier transform arbitrarily well. We also show that the catalyst state can be prepared in time polynomial in $\log (1\mathrm{/}\epsilon )$.