Interplay of measurement and control induced phase transitions in β-adic Rényi circuits

Developing a general understanding of phase transitions far from equilibrium remains a pressing open question in quantum statistical mechanics. Recently, a great deal of attention has focused on measurement-induced phase transitions (MIPT) in the entanglement of the quantum state that occurs due to the competition between entangling unitary dynamics and disentangling random local measurements. Incorporating feedback from the measurement outcome can steer the dynamics to a predetermined final unentangled state, but only after a critical rate of feedback has been applied, defining control-induced phase transitions (CIPTs). The interplay of the MIPT and the CIPT is studied in a variety of Haar random quantum circuits derived from β-adic Rényi maps with different control protocols to understand what details dictate the overall topology of the phase diagram. Several computational approaches are used, and their efficiency will be discussed in detail.