Analytical Nuclear Gradients and Hessians on Quantum Hardware via Orbital-Optimized VQE with Error Mitigation

Abstract

Nuclear gradients and Hessians are fundamental quantities in computational chemistry, essential for a wide range of applications including geometry optimization, vibrational spectroscopy, and molecular property calculations. In this work, we present their analytical implementation on quantum hardware. The methodology is formulated within an active-space framework combining orbital optimization and linear-response theory. On the quantum-computing side, the approach employs the tiled unitary product state (tUPS) ansatz to directly evaluate the tensor elements required for solving the response equations. Moreover, the expectation values are corrected using an adapted confusion-matrix error-mitigation scheme in combination with post-selection criteria. The resulting workflow is assessed on molecular hydrogen and on water through the calculation of potential energy surfaces, nuclear gradients, Hessians, and vibrational frequencies, enabling the evaluation of both its capabilities and current limitations. The results demonstrate good performance for the hydrogen molecule, whereas the water molecule provides a more demanding test of quantum-hardware resources and highlights the trade-offs associated with error-mitigation strategies. The quantified analysis of the results identify the main sources of errors, suggesting improvement directions for more accurate quantum computer applications.