Thermalization and Many‑Body Zeno Effect in Monitored Hamiltonian Dynamics

Random quantum states are essential for quantum information science, with applications ranging from quantum computing to cryptography. Prior approaches for generating these states often rely on using a large bath to thermalize a smaller system, with a subsequent measurement on the bath used to post-select a random state. 

To reduce the required size of the bath, we propose a resource-efficient scheme using holographic deep thermalization driven by Hamiltonian evolution, combined with mid-circuit measurements. By trading spatial and temporal resources, our approach achieves genuine randomness with only a constant-size bath. We quantify the randomness using the frame potential and derive its asymptotic behavior, which shows good agreement with our numerical simulations. Given a total evolution time, as we increase the number of mid-circuit measurements, the frame potential initially decreases exponentially with the number of measurements, due to the mechanism of holographic deep thermalization. Past a critical number of mid‑circuit measurements, the frame potential rises again, signaling the onset of the quantum Zeno effect. Our findings offer practical guidance for generating Haar-random ensembles through Hamiltonian evolution and controlled measurement.