Communications of the ACM
Learning DNF over the uniform distribution using a quantum example oracle
COLT '95 Proceedings of the eighth annual conference on Computational learning theory
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Exponential separation of quantum and classical one-way communication complexity
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Equivalences and Separations Between Quantum and Classical Learnability
SIAM Journal on Computing
Limitations of Quantum Advice and One-Way Communication
CCC '04 Proceedings of the 19th IEEE Annual Conference on Computational Complexity
Exponential lower bound for 2-query locally decodable codes via a quantum argument
Journal of Computer and System Sciences - Special issue: STOC 2003
Bounded-error quantum state identification and exponential separations in communication complexity
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
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We define a new model of quantum learning that we call Predictive Quantum (PQ). This is a quantum analogue of PAC, where during the testing phase the student is only required to answer a polynomial number of testing queries. We demonstrate a relational concept class that is efficiently learnable in PQ, while in any "reasonable" classical model exponential amount of training data would be required. This is the first unconditional separation between quantum and classical learning. We show that our separation is the best possible in several ways; in particular, there is no analogous result for a functional class, as well as for several weaker versions of quantum learning. In order to demonstrate tightness of our separation we consider a special case of one-way communication that we call single-input mode, where Bob receives no input. Somewhat surprisingly, this setting becomes nontrivial when relational communication tasks are considered. In particular, any problem with two-sided input can be transformed into a single-input relational problem of equal classical one-way cost. We show that the situation is different in the quantum case, where the same transformation can make the communication complexity exponentially larger. This happens if and only if the original problem has exponential gap between quantum and classical one-way communication costs. We believe that these auxiliary results might be of independent interest.