Efficient calculation of energy derivatives on a Fault-Tolerant Quantum Computer

Energy derivatives underpin many fundamental properties of molecular systems, from the dipole moments to the hyperfine couplings and forces.

Here, we present new algorithms for efficiently calculating energy derivatives on fault-tolerant quantum computers, exploiting existing state-of-the-art techniques, including block encoding of fermionic operators, use of higher order finite difference formula, and Heisenberg-limited expectation value estimation methods. We optimize the algorithms' parameters to reduce their computational cost and discuss their asymptotic scalings. We show how these approaches can achieve Heisenberg's limited scaling of the errors and compare their different performance, supporting the results with numerical simulations. We will discuss the limits of such techniques and their direct dependence on the cost of state preparation and the calculation of the expectation value of the energy. We will finally explore their applicability to problems of practical relevance, such as the geometry optimization of molecules.