When Geometry Meets Energy: Robust Operator Selection for Adaptive VQE via GAGrad

Adaptive variational quantum eigensolvers (VQE) grow quantum circuits by iteratively selecting operators from a predefined pool. The standard energy-gradient criterion (ADAPT-VQE) aligns directly with the optimization objective but can mislead when the reference state deviates from Hartree-Fock. Geometry-driven alternatives based on the quantum geometric tensor (QGT) offer complementary strengths, prioritizing structurally non-redundant operators and maintaining robustness to reference-state perturbation. We propose the GAGrad operator selection framework, which combines energy-gradient magnitudes with geometric volume-expansion criteria measured via the Schur complement of the QGT. Across molecular benchmarks (LiH, BeH${}_2$; 6–10 qubits), we uncover a gradient failure phenomenon: under moderate perturbation, gradient and Schur-complement rankings become anti-correlated ($\tau =-0.43$), and ADAPT and GAG diverge only in the final 1–2 operator selections, yet this late-stage difference produces 3–4 orders of magnitude gap in residual energy. Thus, geometric information is most consequential near convergence, when energetic differences between candidates become small and gradient rankings become unreliable. GAGrad matches ADAPT's step efficiency while capturing this geometric advantage, achieving comparable precision on most benchmarks with shallower circuits and stable performance under depolarizing noise. A GAG warmstart protocol further reduces circuit depth substantially.