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

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.. Authors: Ragone, Michael [Univ. of California, Davis, CA (United States)]; Bakalov, Bojko N. [North Carolina State University, Raleigh, NC (United States)] (ORCID:0000000346306120); Sauvage, Frédéric Antoine [Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)] (ORCID:0000000333635929); Kemper, Alexander F. [North Carolina State University, Raleigh, NC (United States)] (ORCID:0000000254265181); Ortiz Marrero, Carlos [Pacific Northwest National Laboratory (PNNL), Richland, WA (United States); North Carolina State University, Raleigh, NC (United States)]. DOE Contract: 89233218CNA000001. Subjects: 97 MATHEMATICS AND COMPUTING; barren plateaus; deep parametrized quantum circuit; quantum neural network