  The Grover algorithm is a crucial solution for addressing unstructured search
problems and has emerged as an essential quantum subroutine in various complex
algorithms. This research extensively investigates Grover's search methodology
within non-uniformly distributed databases, a scenario frequently encountered
in practical applications. Our analysis reveals that the behavior of the Grover
evolution differs significantly when applied to non-uniform databases compared
to uniform or 'unstructured databases'. It is observed that the search process
facilitated by this evolution does not consistently result in a speed-up, and
we have identified specific criteria for such situations. Furthermore, we have
extended this investigation to databases characterized by coherent states,
confirming the speed-up achieved through Grover evolution via rigorous
numerical verification. In conclusion, our study provides an enhancement to the
original Grover algorithm, offering insights to optimize implementation
strategies and broaden its range of applications.
