Example of Grover's Algorithm applied to a decision problem?

Question

I recently asked this question on Grover's algorithm, and I am still fairly confused about the whole thing. Consider the following snippet from this post (written by DIDIx13) which for convenience I will reproduce here:

If you throw away the problem structure, and just consider the space of $2^n$ possible solutions, then even a quantum computer needs about $\sqrt{2^n}$ steps to find the correct one (using Grover's algorithm) If a quantum polynomial time algorithm for a $\text{NP}$-complete problem is ever found, it must exploit the problem structure in some way.

The first line emphasis one place where I am confused: Grover's algorithm finds a solution amongst $2^n$ solutions to a problem - this is not a decisions problem alone and as mentioned in my question linked above means we cannot assign it a complexity class.

That said Grover's algorithm can be used to solve decision problems (there seems to be a lot of talk on related questions about "SAT") - but I have yet seen a simple example of such an application.

Thus my question is: Does there exist a simple example of Grover's algorithm solving a decision problem? (even better if you can provide one where the classical search is in $NP$ and another in $P$)

Answer 1

Take the problem of 3-SAT. There is some $f(x)$ which gives outputs 0 or 1. We generally think of the case where the outputs 1 are rare, and hard to find.

PROBLEM: Determine if there is an $x$ that satisfies $f(x)=1$.

(3-SAT has a certain structure to the way the variables are evaluated based on conjunctive normal form, but that's not so important right now).

This problem is known to be NP-complete: as much as we believe NP and P are distinct, we believe this problem cannot be solved efficiently. But Grover's does help us because it gives us a square root speed-up; it searches for the items where the answer is 1.

I'm not sure if you're asking for a specific example of an $f(x)$, but one of the issues is that this complexity classification is about scaling: you need a family of functions for different sized inputs. 3-SAT is one such class. Perhaps my answer here supplies the sorts of examples you were after?