Physics Departament, Center for Natural and Exact Sciences, Federal University of Santa Maria, Roraima Avenue 1000, 97105-900, Santa Maria, RS, Brazil.
Sci Rep. 2023 Jun 20;13(1):9978. doi: 10.1038/s41598-023-37003-5.
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.
尽管我们目前正处于嘈杂的中等规模量子设备时代，但仍有几项研究旨在将机器学习引入量子领域。目前，量子变分电路是构建此类模型的主要策略之一。然而，尽管它被广泛应用，但我们仍然不知道创建量子机器学习模型所需的最少资源是什么。在本文中，我们分析了参数化的表达能力如何影响代价函数。我们从理论上证明，参数化的表达能力越强，代价函数就越倾向于集中在一个值上，这个值不仅取决于所选择的可观测量，还取决于所使用的量子比特数。为此，我们首先得到了参数化的表达能力和代价函数的平均值之间的关系。之后，我们将参数化的表达能力与代价函数的方差联系起来。最后，我们展示了一些数值模拟结果，证实了我们的理论分析预测。据我们所知，这是首次明确地将量子神经网络的这两个重要方面联系起来。
