Although we are currently in the era of noisy intermediate scale quantum devices, several studies are being conducted with the aim of bringing machine learning to the quantum domain. Currently, quantum variational circuits are one of the main strategies used to build such models. However, despite its widespread use, we still do not know what are the minimum resources needed to create a quantum machine learning model. In this article, we analyze how the expressiveness of the parametrization affects the cost function. We analytically show that the more expressive the parametrization is, the more the cost function will tend to concentrate around a value that depends both on the chosen observable and on the number of qubits used. For this, we initially obtain a relationship between the expressiveness of the parametrization and the mean value of the cost function. Afterwards, we relate the expressivity of the parametrization with the variance of the cost function. Finally, we show some numerical simulation results that confirm our theoretical-analytical predictions. To the best of our knowledge, this is the first time that these two important aspects of quantum neural networks are explicitly connected.

翻译：尽管我们目前正处于噪声中等规模量子设备时代，但众多研究正致力于将机器学习引入量子领域。当前，量子变分电路是构建此类模型的主要策略之一。然而，尽管其应用广泛，我们仍不清楚创建量子机器学习模型所需的最少资源。本文分析了参数化表达能力如何影响代价函数。我们从理论上证明，参数化表达能力越强，代价函数越倾向于集中在一个既依赖于所选可观测量又依赖于所用量子比特数量的值周围。为此，我们首先建立了参数化表达能力与代价函数均值之间的关系。随后，我们将参数化表达能力与代价函数方差联系起来。最后，我们展示了一些数值模拟结果，这些结果验证了我们的理论分析预测。据我们所知，这是首次明确将量子神经网络的这两个重要方面联系起来。