The field of quantum information takes a pragmatic approach to examining and utilizing quantum mechanics, seeking to obtain rigorous understandings of which information processing tasks can (e.g. quantum computing, communication) or cannot (e.g. no-cloning theorem) be accomplished according to the laws of Nature. At the heart of these desired tasks is the manipulation of various useful quantum features, most prominent examples being entanglement and coherence, which emerge as valuable “resources” that are needed to empower advantages over classical methods. In practice, a particularly important and widely-studied kind of manipulation is to “purify” the quantum resources, since they are inevitably contaminated by noises and thus often lost their power or become unreliable for direct usage. We prove fundamental limitations on how effectively generic noisy resources can be purified enforced by quantum mechanics, which universally apply to any reasonable kind of quantum resource. Remarkably, it is impossible to achieve perfect resource purification, even probabilistically. Our theorems indicate strong limits on the efficiency of distillation, a widely-used type of resource purification subroutine that underpins many key applications of quantum information science. In particular, we give explicit lower bounds on the cost of magic state distillation, a leading scheme for realizing scalable fault-tolerant quantum computing. This is the first rigorous understanding of the cost required for practical quantum computing to our knowledge.