Dynamics of Loschmidt echoes from operator growth in noisy quantum many-body systems

Abstract: We study the dynamics of Loschmidt echoes in noisy quantum many-body systems without conservation laws. We first show that the operator Loschmidt echo in noisy unitary dynamics is equivalent to the operator norm of the corresponding dissipative dynamics upon noise averaging. We then analyze this quantity in two complementary ways, revealing universal dynamical behavior. First, we develop a heuristic picture for generic Floquet systems that connects Loschmidt echoes, out-of-time-order correlators, and operator growth, which is valid at any dissipation strength. We assert that the Loschmidt echo has two dynamical regimes depending on the time $t$ and the strength of the noise $p$: Gaussian decay for $pt\ll1$ and exponential decay (with a noise-independent decay rate) for $pt\gg1$. Lastly, we rigorously prove all our results for a solvable chaotic many-body quantum circuit, the dissipative random phase model -- thus providing exact insight into dissipative quantum chaos.