Universality of stochastic control of quantum chaos with measurement and feedback

In classical systems governed by unstable fixed points, control can help stabilize the dynamics. Quantization enriches this behavior. We explore this interplay in the paradigmatic case of the inverted harmonic oscillator (IHO), a model whose operator amplitudes grow unboundedly, thereby amplifying the competition between instability and control. We implement a quantum analog of classical stochastic control by combining weak reset with the unstable evolution of the IHO, turning the classically unstable fixed point into a global attractor. We examine how quantum effects directly impact the controlled phase and the dynamics. Using numerical simulation, a semiclassical Fokker-Planck description, and a direct analysis of the quantum channel's spectrum, we identify a sharp transition in the long-time behavior of the system: between a control-dominated regime in which the state is localized about the fixed point and an instability-dominated regime characterized by a runaway divergence of observables. We further extract signatures of purely quantum phenomena, including deviations from the semiclassical picture and non-Gaussian tails in the state distribution, and we argue for universality beyond this toy model. These results point toward a new perspective on probabilistic quantum control in unstable (and potentially chaotic) settings, with implications for measurement-driven phase transitions.