Barren plateaus, complex entanglement, and structured circuits in variational quantum algorithms

Variational Quantum Algorithms (VQAs) offer a promising route for simulating quantum many-body systems on near-term hardware, but their scalability is limited by barren plateaus, where gradients vanish and optimization becomes infeasible. While entanglement growth has been linked to this phenomenon, a quantitative diagnostic connecting circuit structure, entanglement complexity, and trainability remains incomplete. Here we investigate the Cluster–Ising model and the Toric Code Hamiltonian by tracking the entanglement spectrum through r-ratio statistics. We observe that deep circuits tend toward Wigner–Dyson–like behavior, signaling the emergence of complex entanglement that correlates with barren plateau onset. By comparing different circuit geometries—Finite Local Depth Circuits (FLDC), Global Local Depth Circuits (GLDC), and brickwall layouts with Cartan blocks and single-qubit rotation gates—we identify regimes where shallow circuits, including both FLDC and GLDC, can obtain low variational energies exhibiting distinct entanglement-spectrum features. These results suggest that r-ratio diagnostics can help characterize trainability and provide guidance for designing more robust ansätze in near-term VQAs.