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Traditional quantum state estimation requires a number of measurements that grows exponentially with the number of qubits n. But using ideas from computational learning theory, I'll show that "for most practical purposes" one can learn a quantum state using a number of measurements that grows only linearly with n. Besides possible implications for experimental physics, this learning theorem has two applications to quantum computing: first, a new simulation of quantum protocols, and second, the use of trusted classical advice to verify untrusted quantum advice. No quantum computing background is required.