Efficient quantum computation of molecular forces and other energy gradients

While most work on the quantum simulation of chemistry has focused computing energy surfaces, a similarly important application requiring subtly different algorithms is computation derivatives. Almost all molecular properties can be expressed an derivative, including forces, which are essential for applications such as dynamics simulations. Here, we introduce new derivatives with significantly lower complexity than prior methods. Under cost models appropriate noisy-intermediate scale devices, demonstrate how low-rank factorization and other tomography schemes optimized derivative calculations. We numerically that our techniques reduce number circuit repetitions required by many orders magnitude even modest systems, estimating entire force vector may in some systems energy. In context fault-tolerant algorithms, develop methods Heisenberg limited scaling, incorporating state-of-the-art block encoding fermionic operators. contrast to near-term results, find forces any Heisenberg-limited bounded energies, due inner loops either estimation or reflections around ground state. This implies geometry optimization, coupling parameter estimation, spectral prediction practical but tractable simulations large-scale requires further algorithmic advances.