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 could realize the extensively studied entanglement and purification phase transitions, as well as a rich variety of symmetry-protected and ordered non-equilibrium phases. The central question regarding such phases is whether they survive in real-world devices exhibiting unavoidable symmetry-breaking noise. We study the fate of the symmetry-protected absorbing state and charge-sharpening transitions in the presence of symmetry-breaking noise, and establish that the net effect of noise results in coherent and incoherent symmetry-breaking effects. The coherent contribution removes a sharp distinction between different phases and renders phase transitions to crossovers. Nevertheless, states far away from the original phase boundaries retain their essential character. In fact, corrective feedback in adaptive quantum circuits and postselected measurements in symmetric charge-conserving quantum circuits can suppress the effects of noise, thereby stabilizing the absorbing and charge-sharp phases, respectively. Despite the unavoidable noise in current quantum hardwares, our findings offer an optimistic outlook for observing symmetry-protected phases in currently available Noisy Intermediate-Scale Quantum (NISQ) devices. Moreover, our work suggests a symmetry-based benchmarking method as an alternative for characterizing noise and evaluating average local gate fidelity.