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Lower Bounds of Quantum Search for Extreme Point

Yuri Ozhigov
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Abstract

We show that Durr-Hoyer's quantum algorithm of searching for extreme point of integer function can not be sped up for functions chosen randomly. Any other algorithm acting in substantially shorter time $o(\sqrt{2^n})$ gives incorrect answer for the functions with the single point of maximum chosen randomly with probability converging to 1. The lower bound as $Ω(\sqrt{2^n /b})$ was established for the quantum search for solution of equations $f(x)=1$ where $f$ is a Boolean function with $b$ such solutions chosen at random with probability converging to 1.

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Yuri Ozhigov. 1998-12-18. Lower Bounds of Quantum Search for Extreme Point. https://doi.org/10.1098/rspa.1999.0397

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