We explore the important task of applying a phase exp(i f (x)) to a computational basis state |x). The closely related task of rotating a target qubit by an angle depending on f (x) is also studied. Such operations are key in many quantum subroutines, and frequently f (x) can be well approximated by a piecewise function; examples range from the application of diagonal Hamiltonian terms (such as the Coulomb interaction) in grid-based many-body simulation to derivative pricing algorithms. Here, we exploit a parallelization of the piecewise approach so that all constituent elementary rotations are performed simultaneously, that is, we achieve a total rotation depth of one. Moreover, we explore the use of recursive catalyst "towers" to implement these elementary rotations efficiently. We find that strategies prioritizing execution speed can achieve circuit depth as low as O(log2 n+ log2 S) for a register of n qubits and a piecewise approximation of S sections (presuming prior preparation of enabling resource states), albeit total qubit count then scales with S. In the limit of multiple repetitions of the oracle, we find that catalyst tower approaches have an O(S n) T-count.
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