  We introduce and explore a one-dimensional "hybrid" quantum circuit model
consisting of both unitary gates and projective measurements. While the unitary
gates are drawn from a random distribution and act uniformly in the circuit,
the measurements are made at random positions and times throughout the system.
By varying the measurement rate we can tune between the volume law entangled
phase for the random unitary circuit model (no measurements) and a "quantum
Zeno phase" where strong measurements suppress the entanglement growth to
saturate in an area-law. Extensive numerical simulations of the quantum
trajectories of the many-particle wavefunctions (exploiting Clifford circuitry
to access systems up to 512 qubits) provide evidence for a stable "weak
measurement phase" that exhibits volume-law entanglement entropy, with a
coefficient decreasing with increasing measurement rate. We also present
evidence for a novel continuous quantum dynamical phase transition between the
"weak measurement phase" and the "quantum Zeno phase", driven by a competition
between the entangling tendencies of unitary evolution and the disentangling
tendencies of projective measurements. Detailed steady-state and dynamic
critical properties of this novel quantum entanglement transition are accessed.
