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

Quantum 10, 2191 (2026).

https://doi.org/10.22331/q-2026-08-13-2191

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 $log(1/epsilon)$. This implies that $mathsf{QNC}^0_f/mathsf{qpoly}$ 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 $log (1/epsilon)$.