Estimating the ground state energy of the Schrödinger equation for convex potentials

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Abstract In 2011, the fundamental gap conjecture for Schrodinger operators was proven. This can be used to estimate the ground state energy of the time-independent Schrodinger equation with a convex potential and relative error e . 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 e . The cost of the algorithm is polynomial in d and e − 1 , while the number of qubits is polynomial in d and log e − 1 . In addition, we present an algorithm for preparing a quantum state that overlaps within 1 − δ , δ ∈ ( 0 , 1 ) , with the ground state eigenvector of the discretized Hamiltonian. This algorithm also approximates the ground state with relative error e . The cost of the algorithm is polynomial in d , e − 1 and δ − 1 , while the number of qubits is polynomial in d , log e − 1 and log δ − 1 .