Complexity phase transition in non-unitary dynamics

Complexity is among one of the central questions in both quantum information and quantum computation physics. However, the impact of non-unitary dynamics on complexity remains insufficiently understood from an analytical perspective. In this paper, we consider the complexity of a Brownian SYK model with weak measurements. By investigating the frame potential, we analytically show that the weak measurement leads to a complexity phase transition at the same location as the entanglement transition. Further, we study an all-to-all random circuit model under weak measurements by mapping it to a local tree graph. The tree graph may undergo a geometrical transition as the weak measurement exceeds a critical point, corresponding to a complexity phase transition. Numerical simulations of the two models are performed to verity the complexity phase transition, consistent with our analytical results.