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Grover's search algorithm is an algorithm that can perform a search in the order of square root of the input size. This is a provable speed up over the best classical algorithm, which requires a time of order N to perform a search.

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Grover search with single query to $f(x)$ and multiple queries to $f(x) = a$

Assume $f(x)$ is an $n$ bits to $m$ bits function and we want to use Grover's search algorithm to find $x$ such that $f(x) = a$, where $a$ is some $m$-bit predetermined value. When using the algorit …