  Variational algorithms have particular relevance for near-term quantum
computers but require non-trivial parameter optimisations. Here we propose
Analytic Descent: Given that the energy landscape must have a certain simple
form in the local region around any reference point, it can be efficiently
approximated in its entirety by a classical model -- we support these
observations with rigorous, complexity-theoretic arguments. One can classically
analyse this approximate function in order to directly `jump' to the
(estimated) minimum, before determining a more refined function if necessary.
We derive an optimal measurement strategy and generally prove that the
asymptotic resource cost of a `jump' corresponds to only a single gradient
vector evaluation.
