The Adjoint Is All You Need: Characterizing Barren Plateaus in Quantum Ansätze

• DOI: 10.48550/arxiv.2309.07902
• Publication Date: 2023-01-01
• Authors: Enrico Fontana; Dylan Herman; Shouvanik Chakrab...; Niraj Kumar; Romina Yalovetzky; Jamie Heredge; Shree Hari Suresh...; Marco Pistoia
• Keywords: Quantum Physics; FOS: Physical sciences; Quantum Physics (quant-ph)

Using tools from the representation theory of compact Lie groups, we formulate a Barren Plateaus (BPs) for parameterized quantum circuits whose observables lie in their dynamical algebra (DLA), setting that term Supported Ansatz (LASA). A large variety commonly used ans\"atze such as Hamiltonian Variational Ansatz, Quantum Alternating Operator and many equivariant neural networks are LASAs. In particular, our provides, first time, ability to compute variance gradient cost function compound ansatz. We rigorously prove that, LASA, function, 2-design group, scales inversely with dimension DLA, which agrees existing numerical observations. addition, motivate applicability results 2-designs practical settings, show rapid mixing occurs LASAs polynomial DLA. Lastly, include potential extensions handling cases when observable lies outside DLA implications results.