Measurement-induced entanglement phase transitions in variational quantum circuits

Recent studies on measurement-induced entanglement phase transitions in random quantum circuits hint at universal critical behavior. Currently, it is largely unknown if similar phase transitions occur in more structured circuits, e.g., circuits performing Hamiltonian dynamics, since it is difficult to simulate large circuits efficiently and extract the relevant critical behavior. Moreover, the practical applications of these ideas are still largely unexplored. In this work, we show that measurement-induced entanglement phase transitions take place in two prototypical variational quantum circuits, the Hamiltonian variational ansatz (HVA) for the XXZ model and the Hardware efficient ansatz (HEA). We find that the measurement-induced entanglement transition in these systems belongs to the same universality class as found in previous work. In addition, we show that the measurement-induced entanglement transition coincides with a ``landscape transition'' where we observe mild/no barren plateaus in the area law phase of the model. In producing these results, we derived a parameter shift rule for calculating quantum gradients on circuits with intermediate projective measurements, which might be of independent interest.