The Learnability of Quantum States

Abstract

Traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n. But using ideas from computational learning theory, we show that "for most practical purposes" one can learn a state using a number of measurements that grows only linearly with n. Besides possible implications for experimental physics, our learning theorem has two applications to quantum computing: first, a new simulation of quantum one-way communication protocols, and second, the use of trusted classical advice to verify untrusted quantum advice.

30 pages; added discussion of adaptive measurements, moved proofs to appendix, and corrected various minor errors