If Grover algorithm has a better speed than classical search algorithms, would it be an example of where Quantum computers outruns classical computers?
Can we use Grover Algorithm in real world problems?
A Grover algorithm outperform classical unordered database search algorithms quadratically. So, it can serve as an example of higher performance of quantum computers. However, when complexity of Grover search is assessed, generally a complexity of an oracle is ignored. In some cases the oracle complexity is so high that it cancels out advantage of faster search.
Currently, Grover search cannot be used for real world problems because of high noise level and decoherence quantum computers suffer from. However, once these problems are solved, Grover search can be employed naturally for unordered database search and optimization tasks.