Classical Verification of Quantum Learning
Matthias C. Caro, Marcel Hinsche, Marios Ioannou, Alexander Nietner, Ryan Sweke
TL;DR
This work studies how classical verifiers can leverage untrusted quantum servers to solve agnostic learning tasks with quantum data. It introduces mixture-of-superpositions as a flexible quantum data resource that enables distributional agnostic learning and, crucially, distributional Fourier sampling, which supports efficient learning of parities and Fourier-sparse functions. The authors develop interactive verification protocols in which a classical verifier, given random examples or statistical queries, can validate the quantum prover’s outputs, achieving completeness and soundness results and highlighting separations from classical capabilities. They further show that, while mixture-of-superpositions can outperform classical data for distributional learning, they do not provide a broad advantage for distribution-independent agnostic learning or its verification. Overall, the paper demonstrates that quantum data can broaden the set of learnable tasks for classical agents via interaction with quantum servers, while also delineating concrete limitations and the boundary of quantum advantage in delegated learning scenarios.
Abstract
Quantum data access and quantum processing can make certain classically intractable learning tasks feasible. However, quantum capabilities will only be available to a select few in the near future. Thus, reliable schemes that allow classical clients to delegate learning to untrusted quantum servers are required to facilitate widespread access to quantum learning advantages. Building on a recently introduced framework of interactive proof systems for classical machine learning, we develop a framework for classical verification of quantum learning. We exhibit learning problems that a classical learner cannot efficiently solve on their own, but that they can efficiently and reliably solve when interacting with an untrusted quantum prover. Concretely, we consider the problems of agnostic learning parities and Fourier-sparse functions with respect to distributions with uniform input marginal. We propose a new quantum data access model that we call "mixture-of-superpositions" quantum examples, based on which we give efficient quantum learning algorithms for these tasks. Moreover, we prove that agnostic quantum parity and Fourier-sparse learning can be efficiently verified by a classical verifier with only random example or statistical query access. Finally, we showcase two general scenarios in learning and verification in which quantum mixture-of-superpositions examples do not lead to sample complexity improvements over classical data. Our results demonstrate that the potential power of quantum data for learning tasks, while not unlimited, can be utilized by classical agents through interaction with untrusted quantum entities.