  Optimization is one of the keystones of modern science and engineering. Its
applications in quantum technology and machine learning helped nurture
variational quantum algorithms and generative AI respectively. We propose a
general approach to design variational optimization algorithms based on
generative models: the Variational Generative Optimization Network (VGON). To
demonstrate its broad applicability, we apply VGON to three quantum tasks:
finding the best state in an entanglement-detection protocol, finding the
ground state of a 1D quantum spin model with variational quantum circuits, and
generating degenerate ground states of many-body quantum Hamiltonians. For the
first task, VGON greatly reduces the optimization time compared to stochastic
gradient descent while generating nearly optimal quantum states. For the second
task, VGON alleviates the barren plateau problem in variational quantum
circuits. For the final task, VGON can identify the degenerate ground state
spaces after a single stage of training and generate a variety of states
therein.
