  Despite the promise that fault-tolerant quantum computers can efficiently
solve classically intractable problems, it remains a major challenge to find
quantum algorithms that may reach computational advantage in the present era of
noisy, small-scale quantum hardware. Thus, there is substantial ongoing effort
to create new quantum algorithms (or adapt existing ones) to accommodate depth
and space restrictions. By adopting a hybrid query perspective, we identify and
characterize two methods of ``breaking down'' quantum algorithms into rounds of
lower (query) depth, designating these approaches as ``parallelization'' and
``interpolation''. To the best of our knowledge, these had not been explicitly
identified and compared side-by-side, although one can find instances of them
in the literature. We apply them to two problems with known quantum speedup:
calculating the $k$-threshold function and computing a NAND tree. We show that
for the first problem parallelization offers the best performance, while for
the second interpolation is the better choice. This illustrates that no
approach is strictly better than the other, and so that there is more than one
good way to break down a quantum algorithm into a hybrid quantum-classical
algorithm.
