We introduce a quantum Monte Carlo inspired reweighting scheme to accurately compute energies from optimally short quantum circuits. This effectively hybrid quantum-classical approach features both entanglement provided by a short quantum circuit, and the presence of an effective nonunitary operator at the same time. The functional form of this projector is borrowed from classical computation and is able to filter out high-energy components generated by a suboptimal variational quantum heuristic Ansatz. The accuracy of this approach is demonstrated numerically in finding energies of entangled ground states of many-body lattice models. We demonstrate a practical implementation on IBM quantum hardware up to an 8-qubit circuit.