Robustness of quantum chaos and dissipative phase transitions in nonunitary quantum circuits

Random quantum circuits provide an excellent platform to investigate nonequilibrium dynamics of local strongly interacting quantum systems, due to their analytical tractability, simple numerical implementation, and simulability in quantum computing platforms. In particular, the random phase model (RPM) has been used to show the emergence of quantum chaotic features, namely, the ramp in the spectral form factor. However, no system is immune to interactions with an environment or errors in controlling protocols, and, as such, it is of prime importance to test the robustness of dynamical and chaotic features to dissipation. We study the dissipative form factor (DFF) of the RPM with arbitrary one-site nonunitary gates. In the limit of large local dimension, we obtain an exact expression for the DFF averaged over random unitaries. At long times, the system always relaxes (ie the DFF decays) and it undergoes a phase transition at a critical dissipation strength between phases of fast and slow relaxation. Yet, in the slow relaxation phase, the DFF displays a robust ramp for a finite time interval, which can become parametrically larger than the Thouless time and made arbitrarily long. We complement our findings with numerical results for quantum circuits with small local dimension. Besides establishing a regime of robustness of quantum chaotic features in the presence of dissipation, our results pave the way for an understanding of the dynamical content of spectral correlations in non-Hermitian quantum systems.