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

Estimating the ground state energy of the Schr\"odinger equation for convex potentials