Inductive supervised quantum learning
This work provides a foundational link between classical and quantum learning theories, enabling inquiry into standard elements like sample complexity, but it is incremental as it extends classical concepts to quantum domains.
The paper tackles the problem of defining inductive supervised learning in quantum settings by showing that classical definitions coincide due to non-signalling properties, and proves a quantum de Finetti theorem to establish equivalence asymptotically for quantum cases.
In supervised learning, an inductive learning algorithm extracts general rules from observed training instances, then the rules are applied to test instances. We show that this splitting of training and application arises naturally, in the classical setting, from a simple independence requirement with a physical interpretation of being non-signalling. Thus, two seemingly different definitions of inductive learning happen to coincide. This follows from the properties of classical information that break down in the quantum setup. We prove a quantum de Finetti theorem for quantum channels, which shows that in the quantum case, the equivalence holds in the asymptotic setting, that is, for large number of test instances. This reveals a natural analogy between classical learning protocols and their quantum counterparts, justifying a similar treatment, and allowing to inquire about standard elements in computational learning theory, such as structural risk minimization and sample complexity.