Characterizing barren plateaus in quantum ansätze with the adjoint representation

Enrico Fontana, Dylan Herman (), Shouvanik Chakrabarti, Niraj Kumar, Romina Yalovetzky, Jamie Heredge, Shree Hari Sureshbabu and Marco Pistoia
Additional contact information
Enrico Fontana: JPMorganChase
Dylan Herman: JPMorganChase
Shouvanik Chakrabarti: JPMorganChase
Niraj Kumar: JPMorganChase
Romina Yalovetzky: JPMorganChase
Jamie Heredge: JPMorganChase
Shree Hari Sureshbabu: JPMorganChase
Marco Pistoia: JPMorganChase

Nature Communications, 2024, vol. 15, issue 1, 1-12

Abstract: Abstract Variational quantum algorithms, a popular heuristic for near-term quantum computers, utilize parameterized quantum circuits which naturally express Lie groups. It has been postulated that many properties of variational quantum algorithms can be understood by studying their corresponding groups, chief among them the presence of vanishing gradients or barren plateaus, but a theoretical derivation has been lacking. Using tools from the representation theory of compact Lie groups, we formulate a theory of barren plateaus for parameterized quantum circuits whose observables lie in their dynamical Lie algebra, covering a large variety of commonly used ansätze such as the Hamiltonian Variational Ansatz, Quantum Alternating Operator Ansatz, and many equivariant quantum neural networks. Our theory provides, for the first time, the ability to compute the exact variance of the gradient of the cost function of the quantum compound ansatz, under mixing conditions that we prove are commonplace.

Date: 2024
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DOI: 10.1038/s41467-024-49910-w

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