In this paper, we propose a general scheme to analyze the gradient vanishing phenomenon, also known as the barren plateau phenomenon, in training quantum neural networks with the ZX-calculus. More precisely, we extend the barren plateaus theorem from unitary 2-design circuits to any parameterized quantum circuits under certain reasonable assumptions. The main technical contribution of this paper is representing certain integrations as ZX-diagrams and computing them with the ZX-calculus. The method is used to analyze four concrete quantum neural networks with different structures. It is shown that, for the hardware efficient ansatz and the MPS-inspired ansatz, there exist barren plateaus, while for the QCNN ansatz and the tree tensor network ansatz, there exists no barren plateau.
Noise-Induced Barren Plateaus in Variational Quantum Algorithms
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Imperial College London（伦敦帝国学院）；University of Strathclyde（思克莱德大学）；National Physical Laboratory（国家物理实验室）；Center for Nonlinear Studies, Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室非线性研究中心）
Mitigating Barren Plateaus in Quantum Denoising Diffusion Probabilistic Model
School of Computer Science and Technology, University of Science and Technology of China（中国科学技术大学计算机科学与技术学院）；College of Computing and Data Science, Nanyang Technological University（南洋理工大学计算机科学与数据科学学院）；Research Institute of Quantum Technology, The Hong Kong Polytechnic University（香港理工大学量子技术研究所）
Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Theoretical Division, Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室理论部）；Center for Nonlinear Studies, Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室非线性研究中心）
Defeating Barren Plateaus with Task-Aligned Symmetry
Faculty of Information Science and Engineering, Ocean University of China（中国海洋大学信息科学与工程学部）；China Telecom Quantum Information Technology Group Co., LTD.（中国电信量子信息技术集团有限公司）
From barren plateaus through fertile valleys: Conic extensions of parameterised quantum circuits
Institut für Theoretische Physik Leibniz Universität Hannover（汉诺威莱布尼茨大学理论物理研究所）；Institute for Quantum Information RWTH Aachen University（亚琛工业大学量子信息研究所）
Q-Capsule: A Localized Capsule-Based Quantum Neural Architecture for Barren Plateau Mitigation
Florida Atlantic University（佛罗里达大西洋大学）；Khulna University of Engineering & Technology（库尔纳工程与技术大学）；Southern Illinois University Carbondale（南伊利诺伊大学卡本代尔分校）
Does provable absence of barren plateaus imply classical simulability?
Information Sciences, Los Alamos National Laboratory（信息科学，洛斯阿拉莫斯国家实验室）；Quantum Science Center, Oak Ridge（量子科学中心，奥克兰岭）；Theoretical Division, Los Alamos National Laboratory（理论部，洛斯阿拉莫斯国家实验室）；Center for Nonlinear Studies, Los Alamos National Laboratory（非线性研究中心，洛斯阿拉莫斯国家实验室）；University of Strathclyde（斯特拉斯克莱德大学）；Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL)（物理研究所，洛桑联邦理工学院（EPFL））；Donostia International Physics Center（多斯通国际物理中心）；Department of Physics and Astronomy, University of Waterloo（物理与天文学系，滑铁卢大学）；Vector Institute, MaRS Centre（向量研究所，马尔伯里中心）；Complex Systems, Department of Physics, Faculty of Science, Chulalongkorn University（复杂系统，物理系，科学学院，朱拉隆梭大学）；Walter Burke Institute for Theoretical Physics, Caltech（沃尔特·伯克理论物理研究所，加州理工学院）
A Representation-Theoretic Framework for Characterizing Barren Plateaus
International Institute of Physics, Federal University of Rio Grande do Norte（北里奥格兰德联邦大学国际物理研究所）；Instituto de Física de São Carlos, Universidade de São Paulo（圣保罗大学圣卡洛斯物理研究所）
Can Chemically Inspired Parameter Initialization Mitigate Barren Plateaus in Variational Quantum Eigensolvers?
Guangdong Basic Research Center of Excellence for Aggregate Science, School of Science and Engineering, The Chinese University of Hong Kong (Shenzhen)（广东聚合科学卓越研究中心，科学与工程学部，香港中文大学（深圳））；School of Physical and Chemical Sciences, Queen Mary University of London（物理与化学科学学院，伦敦大学玛丽女王学院）
Taming Barren Plateaus in Arbitrary Parameterized Quantum Circuits without Sacrificing Expressibility
Department of Computer Science and Technology, Tsinghua University, Beijing, China（计算机科学与技术系，清华大学，北京，中国）；Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China（叶阳数学科学中心，清华大学，北京，中国）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 100407, China（燕琦湖北京数学科学与应用研究所，北京，中国）；Department of Mathematics, Tsinghua University, Beijing 100084, China（数学系，清华大学，北京，中国）
