Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus

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.

Automated summary

This paper proposes a general scheme to analyze the gradient vanishing phenomenon in training quantum neural networks using the ZX-calculus. By representing integrations as ZX-diagrams and computing them with the ZX-calculus, the barren plateau phenomenon is studied on four concrete quantum neural networks with different structures. It is found that there are barren plateaus for hardware efficient ansatz and MPS-inspired ansatz, while no barren plateau exists for QCNN ansatz and tree tensor network ansatz.