This paper provides a tight analysis of Grover's quantum algorithm for searching in an unsorted database. The authors derive a closed-form formula for the probability of success after any number of iterations of the algorithm, which allows them to determine the number of iterations needed to achieve a high probability of finding the desired element. They also analyze the algorithm's behavior when the target element appears multiple times in the database and present a new algorithm for finding such elements even when the number of solutions is unknown. By combining techniques from Shor's quantum factoring algorithm with Grover's approach, they introduce a new method for approximate quantum counting, enabling the estimation of the number of solutions. The paper also provides a lower bound on the efficiency of any quantum database search algorithm, showing that Grover's algorithm is nearly optimal in terms of the number of table lookups required. The analysis shows that Grover's algorithm requires approximately π/4√N iterations to achieve a high probability of success, and that the number of iterations needed can be reduced to about π/8√N if a 50% success probability is acceptable. The paper also discusses the case when the number of solutions is unknown and presents an algorithm that can find a solution in expected time O(√(N/t)). Finally, the authors provide a lower bound on the number of table lookups required by any quantum algorithm for finding a unique solution, showing that Grover's algorithm is nearly optimal in this regard.This paper provides a tight analysis of Grover's quantum algorithm for searching in an unsorted database. The authors derive a closed-form formula for the probability of success after any number of iterations of the algorithm, which allows them to determine the number of iterations needed to achieve a high probability of finding the desired element. They also analyze the algorithm's behavior when the target element appears multiple times in the database and present a new algorithm for finding such elements even when the number of solutions is unknown. By combining techniques from Shor's quantum factoring algorithm with Grover's approach, they introduce a new method for approximate quantum counting, enabling the estimation of the number of solutions. The paper also provides a lower bound on the efficiency of any quantum database search algorithm, showing that Grover's algorithm is nearly optimal in terms of the number of table lookups required. The analysis shows that Grover's algorithm requires approximately π/4√N iterations to achieve a high probability of success, and that the number of iterations needed can be reduced to about π/8√N if a 50% success probability is acceptable. The paper also discusses the case when the number of solutions is unknown and presents an algorithm that can find a solution in expected time O(√(N/t)). Finally, the authors provide a lower bound on the number of table lookups required by any quantum algorithm for finding a unique solution, showing that Grover's algorithm is nearly optimal in this regard.