Suppressing noise in quantum states is a central challenge in quantum information processing, especially when the resulting states must be reused in subsequent computation. We resolve the two-decade-old open problem of the minimum sample complexity required to transform noisy copies of an arbitrary quantum state into a single higher-fidelity output under unbiased stochastic noise. We formalize this task as quantum purity amplification (QPA), a framework for suppressing noise without relying on problem-specific structure or detailed noise characterization. Compared with previous approaches, our protocol yields exponentially improved sample-complexity scaling with the noise parameter in the strong-noise regime. For depolarized inputs, the optimal protocol is unique, with an asymptotic uniqueness result for general spectra. We further develop two circuit-level implementations of QPA: GQPE–OQPA, a modular and exact implementation of the optimal protocol, and SWAPNET, a tunable-depth, approximate implementation. Numerical simulations of QPA applied to both digital and analog quantum tasks, together with experiments on superconducting quantum processors, demonstrate finite-size realizations and approximations of the optimal QPA protocol.