Barren Plateaus Arrive Before They Are Fatal: Gradient Variance Decay Versus Trainability

Variational quantum machine learning circuits are notoriously hindered by barren plateaus—regions of the parameter space where the variance of the cost function gradient vanishes exponentially with system size (qubit count or ansatz depth). While existing literature treats the mathematical onset of variance decay as an immediate barrier to trainability, we show that trainability persists well past the asymptotic boundary. By analytically tracking state-vector gradients without shot noise alongside finite-sample empirical estimates, we demonstrate that barren plateaus arrive before they are fatal. Specifically, gradient variance decays exponentially, yet gradient descent optimization successfully navigates the landscape for several dimensions beyond the theoretical threshold where variance drops beneath single-shot experimental resolution. This mismatch arises because optimization trajectories exploit low-dimensional effective subspaces rather than isotropic volume. We analyze this transition through the lens of non-convex optimization landscapes, providing refined bounds for practical scaling in near-term quantum hardware.