Abstract: The present invention provides methods and apparatuses for approximating a ground state or a Gibbs state of a k-local Hamiltonian using a quantum computer system. The procedure begins by providing a set of local generalised measurements corresponding to the terms of the k-local Hamiltonian. A set of data qudits is initialised into an initial state and a local generalised measurement is subsequently performed to perturb a subset of the data qudits from an initial state to a perturbed state. The perturbation is accepted or rejected based on a measurement outcome of the local generalised measurement. The perturbation and accept/reject steps are repeated unless or until a stopping condition is met. The result of the procedure is to provably drive the encoded Hamiltonian toward or even into a ground state or a Gibbs state, a useful starting point in many quantum algorithms, such as those relating to materials simulation.