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

TL;DR

This paper introduces a ZX-calculus-based method to analyze the barren plateau phenomenon in quantum neural networks, extending existing theorems and applying them to various architectures to identify when barren plateaus occur.

Contribution

It extends barren plateau analysis to general parameterized quantum circuits using ZX-calculus, providing a new analytical framework for different quantum neural network structures.

Findings

Barren plateaus exist in hardware efficient and MPS-inspired ansatz.

No barren plateau in QCNN and tree tensor network ansatz.

ZX-calculus enables analytical computation of gradients in quantum circuits.

Abstract

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.