Let's say I'd like to search a database and I have a function $f$ such that $f(x) = 0$ for all incorrect entries and $f(x) = 1$ for all target entries.

The crux of Grover's search seems to be that there exists a way to construct a quantum operator $\hat{f}$ in a way that it takes a superposition of states as input and adds a phase only to the target elements from the superposition.

What guarantee do we have that such a construction exists? For example, if I take a classical hashing function, how do I go about building the quantum equivalent that is able to simultaneously evaluate multiple inputs and assign the phase only to those inputs that fall below some target threshold?


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