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The learnability of quantum states. (English) Zbl 1140.81317
Summary: 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 one can do exponentially better in a statistical setting. In particular, to predict the outcomes of most measurements drawn from an arbitrary probability distribution, one needs only a number of sample measurements that grows linearly with \(n\). This theorem has the conceptual implication that quantum states, despite being exponentially long vectors, are nevertheless ‘reasonable’ in a learning theory sense. The theorem also 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.

MSC:
81P68 Quantum computation
94A05 Communication theory
91E40 Memory and learning in psychology
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