Machine learning algorithms based on parametrized quantum circuits are prime candidates for near-term applications on noisy quantum computers. In this direction, various types of quantum machine learning models have been introduced and studied extensively. Yet, our understanding of how these models compare, both mutually and to classical models, remains limited. In this work, we identify a constructive framework that captures all standard models based on parametrized quantum circuits: that of linear quantum models. In particular, we show using tools from quantum information theory how data re-uploading circuits, an apparent outlier of this framework, can be efficiently mapped into the simpler picture of linear models in quantum Hilbert spaces. Furthermore, we analyze the experimentally-relevant resource requirements of these models in terms of qubit number and amount of data needed to learn. Based on recent results from classical machine learning, we prove that linear quantum models must utilize exponentially more qubits than data re-uploading models in order to solve certain learning tasks, while kernel methods additionally require exponentially more data points. Our results provide a more comprehensive view of quantum machine learning models as well as insights on the compatibility of different models with NISQ constraints.

翻译：基于参数化量子电路的机器学习算法是近期噪声量子计算机应用的主要候选方案。在这一方向上，多种类型的量子机器学习模型已被提出并得到广泛研究。然而，我们对这些模型之间以及它们与经典模型的比较理解仍然有限。在本工作中，我们确定了一个能够涵盖所有基于参数化量子电路的标准模型的建构性框架：即线性量子模型。特别地，我们利用量子信息论的工具证明，数据重上传电路（该框架中一个看似特例的模型）可被高效映射到量子希尔伯特空间中更简洁的线性模型图景。此外，我们分析了这些模型在实验相关资源需求方面的特征，包括所需的量子比特数和学习所需的数据量。基于经典机器学习的最新成果，我们证明：为解决特定学习任务，线性量子模型必须使用的量子比特数按指数级多于数据重上传模型，而核方法额外需要指数级更多的数据点。我们的研究为量子机器学习模型提供了更全面的视角，并揭示了不同模型与NISQ约束兼容性的关键洞见。