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

Chen Zhao and Xiao-Shan Gao

Academy of Mathematics and Systems Science, Chinese Academy of Sciences
University of Chinese Academy of Sciences

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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.

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Cited by

[1] M. Bilkis, M. Cerezo, Guillaume Verdon, Patrick J. Coles, and Lukasz Cincio, "A semi-agnostic ansatz with variable structure for quantum machine learning", arXiv:2103.06712.

[2] Bob Coecke, Dominic Horsman, Aleks Kissinger, and Quanlong Wang, "Kindergarden quantum mechanics graduates (...or how I learned to stop gluing LEGO together and love the ZX-calculus)", arXiv:2102.10984.

[3] Alexis Toumi, Richie Yeung, and Giovanni de Felice, "Diagrammatic Differentiation for Quantum Machine Learning", arXiv:2103.07960.

[4] Martin Larocca, Piotr Czarnik, Kunal Sharma, Gopikrishnan Muraleedharan, Patrick J. Coles, and M. Cerezo, "Diagnosing barren plateaus with tools from quantum optimal control", arXiv:2105.14377.

[5] Marcello Benedetti, Brian Coyle, Mattia Fiorentini, Michael Lubasch, and Matthias Rosenkranz, "Variational inference with a quantum computer", arXiv:2103.06720.

[6] Hiroshi C. Watanabe, Rudy Raymond, Yu-ya Ohnishi, Eriko Kaminishi, and Michihiko Sugawara, "Optimizing Parameterized Quantum Circuits with Free-Axis Selection", arXiv:2104.14875.

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