Noise resilience in adaptive and symmetric monitored quantum circuits

Monitored quantum circuits offer great perspectives for exploring the interplay of quantum information and complex quantum dynamics. These systems exhibit entanglement and purification phase transitions, along with various symmetry-enforced and ordered non-equilibrium phases. The central question is whether these phases can persist in real-world noisy devices. We study the fate of the symmetry-enforced absorbing state and charge-sharpening transitions in the presence of noise, and establish that noise results in coherent and incoherent symmetry-breaking effects. The coherent effects blur the distinction between phases, turning sharp transitions into crossovers, but states far from phase boundaries largely retain their essential character. We find, corrective feedback and postselected measurements can mitigate noise, stabilizing the absorbing and charge-sharp phases. Hence, if challenges like postselection are addressed, errors do not prevent the observation of symmetry-enforced phases in noisy intermediate-scale quantum (NISQ) devices. Additionally, we propose a symmetry-based method to characterize noise and gate fidelity.