Entanglement dynamics in Floquet random circuits under cooling cycles

Quantum many-body systems undergoing random unitary evolution typically gravitate toward a maximally mixed state. However, local projective measurements can effectively purify the system state, leading to measurement-induced phase transitions. In this work, we adopt a different approach: instead of projective measurements, we utilize auxiliary qubits that are stochastically coupled to system qubits and periodically reset to their lower entropy initial state throughout the time evolution. We show that this protocol can cool down a random circuit, decreasing its entropy depending on the probability of interaction, the number of auxiliaries, and the Floquet period. We then analytically derive an upper bound to the entanglement entropy of random Floquet circuits by utilizing the temporally random circuits and determine the conditions in Floquet circuits that saturate this bound. Our current findings suggest that this cooling protocol might also purify the circuit state and result in entanglement transitions. Our work also generalizes the cooling protocols based on engineered dissipation to a wide range of setups.