A Lie algebraic theory of barren plateaus for deep parameterized quantum circuits

Michael Ragone, Bojko N. Bakalov, Frédéric Sauvage, Alexander F. Kemper, Carlos Ortiz Marrero, Martín Larocca and M. Cerezo ()
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Michael Ragone: University of California Davis
Bojko N. Bakalov: North Carolina State University
Frédéric Sauvage: Los Alamos National Laboratory
Alexander F. Kemper: North Carolina State University
Carlos Ortiz Marrero: Pacific Northwest National Laboratory
Martín Larocca: Los Alamos National Laboratory
M. Cerezo: Los Alamos National Laboratory

Nature Communications, 2024, vol. 15, issue 1, 1-10

Abstract: Abstract Variational quantum computing schemes train a loss function by sending an initial state through a parametrized quantum circuit, and measuring the expectation value of some operator. Despite their promise, the trainability of these algorithms is hindered by barren plateaus (BPs) induced by the expressiveness of the circuit, the entanglement of the input data, the locality of the observable, or the presence of noise. Up to this point, these sources of BPs have been regarded as independent. In this work, we present a general Lie algebraic theory that provides an exact expression for the variance of the loss function of sufficiently deep parametrized quantum circuits, even in the presence of certain noise models. Our results allow us to understand under one framework all aforementioned sources of BPs. This theoretical leap resolves a standing conjecture about a connection between loss concentration and the dimension of the Lie algebra of the circuit’s generators.

Date: 2024
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DOI: 10.1038/s41467-024-49909-3

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