  We propose an extension of the Variational Quantum Eigensolver (VQE) that
leads to more accurate energy estimations and can be used to study excited
states. The method is based on the introduction of a sequence of increasing
penalties in the cost function. This approach does not require circuit
modifications and thus can be applied with no additional depth cost. Through
numerical simulations, we show that we are able to produce variational states
with desired physical properties, such as total spin and charge. We assess its
performance both on classical simulators and on currently available quantum
devices, calculating the potential energy curves of small molecular systems in
different physical configurations. Finally, we compare our method to the
original VQE and to another extension, obtaining a better agreement with exact
simulations for both energy and targeted physical quantities.
