The Quantum Search Algorithms for All Solutions

Two quantum search algorithms are proposed for known and unknown number of solutions. The first algorithm begins with an arbitrary rotation phase Grover search algorithm by recursive equations, then a sub-algorithm (G _α algorithm) and the corresponding quantum circuits are designed, the probability of success and expected number of iterations of the sub-algorithm to find a solution are analyzed. Based on these results, we design the whole algorithm and estimate the expected number of iterations to search all solutions. The design of the second algorithm follows the same steps. We firstly study a quantum counting algorithm, then design a sub-algorithm (QCG algorithm), and analyze the probability of success and the expected number of iterations to find a solution. Finally the whole algorithm for all solutions is designed and analyzed. The analysis results show that these two algorithms find all solutions in the expected times of $O(t\sqrt{N})$ (t is a number of solutions and N is a total of states).