Symmetry-Protected Basin Localization in Variational Quantum Eigensolvers

Variational quantum eigensolvers fail before optimization begins when strong correlation splits the molecular energy landscape into competing basins and the initial state selects a non-ground-state basin. We introduce a geometry-conditioned preconditioner ${\mathcal{P}}_{\mathrm{e}\mathrm{q}}:\mathbf{R}\mapsto {\mathit{\theta }}_0$ constrained by the $SE(3)$ covariance of the molecular Hamiltonian, so that nuclear geometry is mapped directly into circuit parameters in the correlated ground-state basin. This basin localization changes the relevant gradient statistics from concentration controlled to curvature controlled. In statevector benchmarks on six stretched molecules, ${\mathcal{P}}_{\mathrm{e}\mathrm{q}}$ reduces Hartree–Fock initialization errors by factors of $38\times$–$6250\times$, reaches sub-mHa initialization in CO, LiH, and H${}_8$, and places N${}_2$, H${}_2$O, and BeH${}_2$ in the mHa-scale correlated basin. In disordered H${}_{10}$ chains, equivariant basin targeting and stochastic escape reach unit success probability at fixed optimization budget. The procedure performs basin selection before the shot-limited quantum loop; the quantum circuit then refines correlation inside the selected basin.