Cost function dependent barren plateaus in shallow parametrized quantum circuits

• DOI: 10.1038/s41467-021-21728-w
• Publication Date: 2021-03-19T13:05:27Z
• Authors: M. Cerezo; Akira Sone; Tyler Volkoff; Lukasz Cincio; Patrick J. Coles
• Keywords: FOS: Computer and information sciences; Quantum Physics; Computer Science - Machine Learning; Science; Q; FOS: Physical sciences; 01 natural sciences; Article; Machine Learning (cs.LG); 0103 physical sciences; Quantum Physics (quant-ph)

Abstract Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized circuit V ( ) to minimize cost function C . While VQAs may enable practical applications noisy computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming is an alternating layered ansatz composed blocks forming local 2-designs. Our first result states that defining in terms global observables leads exponentially vanishing gradients (i.e., barren plateaus) even when shallow. Hence, several literature must revise their proposed costs. On other hand, our second at worst polynomially gradient, so long as depth $${\mathcal{O}}(\mathrm{log}\,n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>log</mml:mi> <mml:mspace/> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:math> results establish connection between locality and trainability. We illustrate these ideas large-scale simulations, up 100 qubits, autoencoder implementation.