  In the recent years, we have seen that Grover search algorithm (Proceedings,
28th annual ACM symposium on the theory of computing, pp. 212-219, 1996) by
using quantum parallelism has revolutionized the field of solving huge class of
NP problems in comparisons to classical systems. In this work, we explore the
idea of extending Grover search algorithm to approximate algorithms. Here we
try to analyze the applicability of Grover search to process an unstructured
database with a dynamic selection function in contrast to the static selection
function used in the original work (Grover in Proceedings, 28th annual ACM
symposium on the theory of computing, pp. 212-219, 1996). We show that this
alteration facilitates us to extend the application of Grover search to the
field of randomized search algorithms. Further, we use the dynamic Grover
search algorithm to define the goals for a recommendation system based on which
we propose a recommendation algorithm which uses binomial similarity
distribution space giving us a quadratic speedup over traditional classical
unstructured recommendation systems. Finally, we see how dynamic Grover search
can be used to tackle a wide range of optimization problems where we improve
complexity over existing optimization algorithms.
