  Efficiently controlling linear Gaussian quantum (LGQ) systems is a
significant task in both the study of fundamental quantum theory and the
development of modern quantum technology. Here, we propose a general
quantum-learning-control method for optimally controlling LGQ systems based on
the gradient-descent algorithm. Our approach flexibly designs the loss function
for diverse tasks by utilizing first- and second-order moments that completely
describe the quantum state of LGQ systems. We demonstrate both deep
optomechanical cooling and large optomechanical entanglement using this
approach. Our approach enables the fast and deep ground-state cooling of a
mechanical resonator within a short time, surpassing the limitations of
sideband cooling in the continuous-wave driven strong-coupling regime.
Furthermore, optomechanical entanglement could be generated remarkably fast and
surpass several times the corresponding steady-state entanglement, even when
the thermal phonon occupation reaches one hundred. This work will not only
broaden the application of quantum learning control, but also open an avenue
for optimal control of LGQ systems.
