The seminar will present a quantum algorithm to sample eigenstates from any desired spectral interval. A special case of this algorithm samples ground states of quantum mechanical systems and lattice field theories. More concretely, we will sketch out an explicit quantum circuit that implements the proposed Spectral Sieve using Fourier transformation on operators, given a description of a Hamiltonian. We will then demonstrate how the algorithm can be generally implemented on existing quantum computers and explore the elementary case of one qubit state preparation. Time permitting, we may discuss a conjecture about the computational complexity of this algorithm. The seminar will be based on first principles and assume minimal background.