  In 2011, the fundamental gap conjecture for Schr\"odinger operators was
proven. This can be used to estimate the ground state energy of the
time-independent Schr\"odinger equation with a convex potential and relative
error \epsilon. Classical deterministic algorithms solving this problem have
cost exponential in the number of its degrees of freedom d. We show a quantum
algorithm, that is based on a perturbation method, for estimating the ground
state energy with relative error \epsilon. The cost of the algorithm is
polynomial in d and \epsilon^{-1}, while the number of qubits is polynomial in
d and \log\epsilon^{-1}. In addition, we present an algorithm for preparing a
quantum state that overlaps within 1-\delta, \delta \in (0,1), with the ground
state eigenvector of the discretized Hamiltonian. This algorithm also
approximates the ground state with relative error \epsilon. The cost of the
algorithm is polynomial in d, \epsilon^{-1} and \delta^{-1}, while the number
of qubits is polynomial in d, \log\epsilon^{-1} and \log\delta^{-1}.
