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

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}.