  Algorithms with unitary oracles can be nested, which makes them extremely
versatile. An example is the phase estimation algorithm used in many candidate
algorithms for quantum speed-up. The search for new quantum algorithms benefits
from understanding their limitations: Some tasks are impossible in quantum
circuits, although their classical versions are easy, for example, cloning. An
example with a unitary oracle $U$ is the if clause, the task to implement
controlled $U$ (up to the phase on $U$). In classical computation the
conditional statement is easy and essential. In quantum circuits the if clause
was shown impossible from one query to $U$. Is it possible from polynomially
many queries? Here we unify algorithms with a unitary oracle and develop a
topological method to prove their limitations: No number of queries to $U$ and
$U^\dagger$ lets quantum circuits implement the if clause, even if admitting
approximations, postselection and relaxed causality. We also show limitations
of process tomography, oracle neutralization, and $\sqrt[\dim U]{U}$, $U^T$,
and $U^\dagger$ algorithms. Our results strengthen an advantage of linear
optics, challenge the experiments on relaxed causality, and motivate new
algorithms with many-outcome measurements.
