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

الأصل · EN

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 ε. 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 ε. The cost of the algorithm is polynomial in d and ε⁻¹, while the number of qubits is polynomial in d and ε⁻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 ε. The cost of the algorithm is polynomial in d, ε⁻¹ and δ⁻¹, while the number of qubits is polynomial in d, ε⁻¹ and δ⁻¹.