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 ϵ -approximation, it suffices to have a catalyst state of size polynomial in log ⁡ ( 1 / ϵ ) . This implies that Q N C f 0 / q p o l 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 / ϵ ) .