A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.
翻译:最近开展的大量工作已开始探讨在混合量子古典优化框架内,作为机器学习模型的半称量子电路(PQCs)作为机器学习模型的潜力,特别是,在一般化界限方面,这类模型的超模性能的理论保障已经出现,然而,这些概括性界限没有一项明确取决于如何将典型输入数据编码到PQC。我们为基于PQC的模型得出一般化界限,这些模型明确取决于数据编码的战略。这意味着对基于PQC的、经过培训的、基于不可见数据的模型的性能的界限。此外,我们的成果有助于通过尽量减少结构风险选择最佳数据编码战略,这是一个数学上严格的模型选择框架。我们通过将基于PQC的模型的复杂程度与Rademacher复杂程度以及基于统计学习理论的矩阵的两种复杂度衡量标准结合起来,从而获得我们的普遍化界限。为了实现这一点,我们依靠通过三角测量功能代表基于PQC的模型。此外,我们的成果有助于选择最佳数据编码战略的重要性。