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.

中文翻译：

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参数化量子电路的编码相关泛化边界

最近的大量工作已经开始在混合量子经典优化的框架内探索参数化量子电路 (PQC) 作为机器学习模型的潜力。特别是，在泛化边界方面，已经出现了对此类模型的样本外性能的理论保证。然而，这些泛化边界都没有明确依赖于经典输入数据如何编码到 PQC 中。我们推导出基于 PQC 的模型的泛化边界，这些模型明确依赖于用于数据编码的策略。这些意味着训练有素的基于 PQC 的模型在看不见的数据上的性能界限。此外，我们的结果通过结构风险最小化促进了最佳数据编码策略的选择，这是一个数学上严格的模型选择框架。我们通过限制基于 PQC 的模型的复杂性来获得我们的泛化边界，该复杂性由 Rademacher 复杂性和度量熵衡量，这两个来自统计学习理论的复杂性度量。为了实现这一点，我们依靠三角函数表示基于 PQC 的模型。我们的泛化界限强调了经过深思熟虑的数据编码策略对于基于 PQC 的模型的重要性。