While most work on the quantum simulation of chemistry has focused on
computing energy surfaces, a similarly important application requiring subtly
different algorithms is the computation of energy derivatives. Almost all
molecular properties can be expressed an energy derivative, including molecular
forces, which are essential for applications such as molecular dynamics
simulations. Here, we introduce new quantum algorithms for computing molecular
energy derivatives with significantly lower complexity than prior methods.
Under cost models appropriate for noisy-intermediate scale quantum devices we
demonstrate how low rank factorizations and other tomography schemes can be
optimized for energy derivative calculations. We perform numerics revealing
that our techniques reduce the number of circuit repetitions required by many
orders of magnitude for even modest systems. In the context of fault-tolerant
algorithms, we develop new methods of estimating energy derivatives with
Heisenberg limited scaling incorporating state-of-the-art techniques for block
encoding fermionic operators. Our results suggest that the calculation of
forces on a single nucleus may be of similar cost to estimating energies of
chemical systems, but that further developments are needed for quantum
computers to meaningfully assist with molecular dynamics simulations.Comment: 48 pages, 14 page appendix, 10 figures. v2 contains updated lambdas
(rescaling factors) for sparse FT encodings and some NISQ methods, obtained
by localizing orbital