  Variational quantum algorithms (VQAs) promise efficient use of near-term
quantum computers. However, training VQAs often requires an extensive amount of
time and suffers from the barren plateau problem where the magnitude of the
gradients vanishes with increasing number of qubits. Here, we show how to
optimally train VQAs for learning quantum states. Parameterized quantum
circuits can form Gaussian kernels, which we use to derive adaptive learning
rates for gradient ascent. We introduce the generalized quantum natural
gradient that features stability and optimized movement in parameter space.
Both methods together outperform other optimization routines in training VQAs.
Our methods also excel at numerically optimizing driving protocols for quantum
control problems. The gradients of the VQA do not vanish when the fidelity
between the initial state and the state to be learned is bounded from below. We
identify a VQA for quantum simulation with such a constraint that thus can be
trained free of barren plateaus. Finally, we propose the application of
Gaussian kernels for quantum machine learning.
