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.
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