Efficient Measure for the Expressivity of Variational Quantum Algorithms

• Published in: Physical review letters Vol. 128; no. 8; p. 080506
• Main Authors: Du, Yuxuan, Tu, Zhuozhuo, Yuan, Xiao, Tao, Dacheng
• Format: Journal Article
• Language: English
• Published: United States 25.02.2022
• Subjects: Algorithms; Neural Networks, Computer
• ISSN: 1079-7114
• DOI: 10.1103/PhysRevLett.128.080506

The superiority of variational quantum algorithms (VQAs) such as quantum neural networks (QNNs) and variational quantum eigensolvers (VQEs) heavily depends on the expressivity of the employed Ansätze. Namely, a simple Ansatz is insufficient to capture the optimal solution, while an intricate Ansatz leads to the hardness of trainability. Despite its fundamental importance, an effective strategy of measuring the expressivity of VQAs remains largely unknown. Here, we exploit an advanced tool in statistical learning theory, i.e., covering number, to study the expressivity of VQAs. Particularly, we first exhibit how the expressivity of VQAs with an arbitrary Ansätze is upper bounded by the number of quantum gates and the measurement observable. We next explore the expressivity of VQAs on near-term quantum chips, where the system noise is considered. We observe an exponential decay of the expressivity with increasing circuit depth. We also utilize the achieved expressivity to analyze the generalization of QNNs and the accuracy of VQE. We numerically verify our theory employing VQAs with different levels of expressivity. Our Letter opens the avenue for quantitative understanding of the expressivity of VQAs.