As I understand Grover's algorithm, given the output of a black-box function, can be used to find the corresponding input (or set of inputs if the function is not one-to-one). It is therefore potentially useful for solving inversion problems that are hard to solve/invert directly but are easy to verify.

Two areas of math that fit that criteria are zero-finding and differential equation solving, i.e. solving a differential equation may be difficult, but checking whether a proposed solution is correct is relatively easy. However, I am unable to find any discussion of whether Grover's algorithm is applicable to this type of problem, which leads me to believe that it's probably not. If that's the case, why not? Is it a fundamental problem, or just that there are already classical algorithms that outperform Grover's?

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    $\begingroup$ Being a bit more precise, Grover's search tests discrete inputs. Solutions to differential equations are not discrete. That said, you might be interested in the HHL algorithm, which can be applied to solving differential equations. $\endgroup$
    – DaftWullie
    May 18, 2021 at 15:48


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