A new measure pinpoints when quantum circuits escape the vanishing gradient trap

Variational quantum algorithms suffer from barren plateaus — regions where gradients shrink exponentially, making training impossible — but most analyses assume idealized random-unitary ensembles that say little about how to choose parameter distributions in practice. The structural f-divergence introduced here quantifies how two distributions on circuit parameters differ in a way directly tied to gradient and cost-function statistics, yielding trade-off inequalities whose bounds are tight (achieved by a single-qubit, single-layer circuit). The framework converts abstract barren-plateau avoidance into concrete necessary and sufficient conditions on the parameter distribution, and also handles noise-induced deviations.