Title: Criteria for Grover Search on Weighted Databases

Abstract: The Grover algorithm stands as a pivotal solution for unstructured search problems and has become a fundamental quantum subroutine in numerous complex algorithms. This study delves into Grover's search methodology within non-uniformly distributed databases, a scenario more commonly encountered in real-world problems. We uncover that in such cases, the Grover evolution displays distinct behavior compared to uniform or 'unstructured databases'. The search enabled by this evolution doesn't consistently yield a speed-up, and we establish criteria for such occurrences. Additionally, we apply this theory to databases whose distributions relate to coherent states, substantiating the speed-up via Grover evolution through numerical verification. Overall, our findings offer an effective extension of the original Grover algorithm, enriching implementation strategies and widening its application scope.