In 2011, the fundamental gap conjecture for Schrödinger operators was proven. This can be used to estimate the ground state energy of the time-independent Schrödinger equation with a convex potential and relative error . Classical deterministic algorithms solving this problem have cost exponential in the number of its degrees of freedom . We show a quantum algorithm, that is based on a perturbation method, for estimating the ground state energy with relative error . The cost of the algorithm is polynomial in and , while the number of qubits is polynomial in and . In addition, we present an algorithm for preparing a quantum state that overlaps within , with the ground state eigenvector of the discretized Hamiltonian. This algorithm also approximates the ground state with relative error . The cost of the algorithm is polynomial in , and , while the number of qubits is polynomial in , and .