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
)
.