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实现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人可用。因此，需要设计可靠的方案，使经典客户端能够将学习任务委托给不可信的量子服务器，从而促进量子学习优势的广泛获取。基于最近提出的经典机器学习交互式证明系统框架，我们构建了一个用于经典验证量子学习的框架。我们展示了经典学习者在没有交互时无法高效解决的学习问题，但当他们与不可信的量子证明者交互时，可以高效且可靠地解决这些问题。具体而言，我们研究了在输入边际分布均匀的条件下，无偏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，并基于此给出了这些任务的高效量子学习算法。此外，我们证明经典验证者仅需随机样本或统计查询访问即可高效验证无偏量子奇偶学习和傅里叶稀疏学习。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并未带来比经典数据更优的样本复杂度。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，但经典代理可以通过与不可信量子实体的交互来利用这一能力。