  Variational quantum algorithms (VQAs) promise efficient use of near-term
quantum computers. However, training these algorithms 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 a VQA for learning quantum states. Parameterized
quantum circuits can form Gaussian kernels, which we use to derive optimal
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
and can enhance VQAs as well as quantum control techniques. 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 can be trained free of barren plateaus. Finally, we
propose the application of Gaussian kernels for quantum machine learning.
