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.
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