Measurement-driven entanglement transition in hybrid quantum circuits

In this paper we continue to explore "hybrid" quantum circuit models in one-dimension with both unitary and measurement gates, focusing on the entanglement properties of wave function trajectories at long times, in the steady state. We simulate a large class of Clifford circuits, including models with or without randomness in the unitary gates, and with or without randomness in the locations of measurement gates, using stabilizer techniques to access the long-time dynamics of systems up to 512 qubits. In all models we find a volume-law entangled phase for low measurement rates, which exhibits a subdominant logarithmic behavior in the entanglement entropy, S-A = alpha ln vertical bar A vertical bar + s vertical bar A vertical bar, with subsystem size vertical bar A vertical bar. With increasing measurement rate the volume-law phase is unstable to a disentangled area-law phase, passing through a single entanglement transition at a critical rate of measurement. At criticality we find a purely logarithmic entanglement entropy, S-A = alpha (p(c)) ln vertical bar A vertical bar, a power-law decay and conformal symmetry of the mutual information, with exponential decay off criticality. Various spin-spin correlation functions also show slow decay at criticality. Critical exponents are consistent across all models, indicative of a single universality class. These results suggest the existence of an effective underlying statistical mechanical model for the entanglement transition. Beyond Clifford circuit models, numerical simulations of up to 20 qubits give consistent results.