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It is well-known that Grover's algorithm is asymptotically optimal. However, is a better algorithm possible by constant factors? And if so, is it known?

Suppose a fixed number of iterations $r$ and ratio $\rho$ of market elements such that, by using the geometric interpretation, the angle does not exceed $\pi/2$ (something like a truncated Grover). For example, for $r = 1$, think in a ratio $\rho \leq 0.25$.

My question is that in such conditions, Grover hits the highest possible probability of measuring a market element (the best amplification) for an quantum algorithm that uses $r$ calls of the oracle.

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Grover's search is exactly (not just asymptotically) optimal: https://arxiv.org/abs/quant-ph/9711070

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