Quantum Mechanical Algorithms for Solving Root Finding Problem

Root finding problem is a hard problem in computational number theory. To improve the efficiency of solving the problem and expand the scope of applications of quantum computing,the quantum algorithms for solving the root finding problem are analyzed in this paper. Based on the two important quantum algorithms—Shor's algorithm and Grover's algorithm, two quantum mechanical algorithms are proposed to solve the root finding problem,which are named RF-Shor algorithm and RFGrover algorithm. According to the analysis,the RF-Shor algorithm can find all the solutions to the root finding problem with a probability closed to 100%,with a consumption of polynomial quantum gates; the RF-Grover algorithm does not involve any classical method to improve the search efficiency and one of the k solutions to the root finding problem can be obtained in O($\sqrt{M/k}$) steps with a probability of at least 50%.