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：大量近期工作已开始探索参数化量子电路（PQC）作为机器学习模型在混合量子-经典优化框架中的潜力。具体而言，这类模型在样本外性能方面的理论保证（即泛化界）已崭露头角。然而，这些泛化界均未明确依赖经典输入数据如何编码到PQC中。我们推导了基于PQC的模型明确依赖于数据编码策略的泛化界。这些界蕴含了训练后的PQC模型在未见数据上的性能保证。此外，我们的结果通过结构风险最小化（一种数学上严格的模型选择框架）实现了最优数据编码策略的选择。我们通过限制基于PQC模型的复杂度来获得泛化界，该复杂度以统计学习理论中的两种度量——拉德马赫复杂度和度量熵——来衡量。为此，我们依赖基于PQC模型的三角函数表示。我们的泛化界强调了精心设计的数据编码策略对于基于PQC模型的重要性。