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.

翻译：量子数据访问与量子处理能够使某些经典上难以处理的学习任务变得可行。然而，在不久的将来，量子能力仅会为少数精英所掌握。因此，需要可靠的方案允许经典客户端将学习任务委托给不可信的量子服务器，以促进对量子学习优势的广泛访问。基于最近提出的用于经典机器学习的交互式证明系统框架，我们开发了用于经典验证量子学习的框架。我们展示了某些学习问题：经典学习者无法独立高效解决，但通过与不可信的量子证明者交互，他们能够高效且可靠地解决这些问题。具体而言，我们考虑在输入边缘分布均匀的分布下，不可知学习奇偶函数和傅里叶稀疏函数的问题。我们提出了一种新的量子数据访问模型，称为“混合叠加”量子样本，并基于该模型给出了这些任务的高效量子学习算法。此外，我们证明了具有随机样本或统计查询访问的经典验证者能够高效验证不可知量子奇偶和傅里叶稀疏学习。最后，我们展示了学习和验证中的两个通用场景，其中量子混合叠加样本在样本复杂度上并未相较于经典数据带来改进。我们的结果表明，量子数据在学习任务中的潜在能力虽非无限，但可通过经典代理与不可信量子实体的交互加以利用。