# Calculating energy derivatives for quantum chemistry on a quantum computer

## TL;DR

This paper introduces two new methods for calculating energy derivatives in quantum chemistry on quantum computers, enabling geometry optimization and property estimation with experimental validation.

## Contribution

It presents novel quantum algorithms for energy derivatives, including quantum phase estimation and response approximation, with experimental implementation on a quantum processor.

## Key findings

- Achieved the first quantum-processor-based geometry optimization of H2.
- Estimated H2 polarizability with 2% relative error.
- Provided error bounds and computational scalings for the methods.

## Abstract

Modeling chemical reactions and complicated molecular systems has been proposed as the `killer application' of a future quantum computer. Accurate calculations of derivatives of molecular eigenenergies are essential towards this end, allowing for geometry optimization, transition state searches, predictions of the response to an applied electric or magnetic field, and molecular dynamics simulations. In this work, we survey methods to calculate energy derivatives, and present two new methods: one based on quantum phase estimation, the other on a low-order response approximation. We calculate asymptotic error bounds and approximate computational scalings for the methods presented. Implementing these methods, we perform the world's first geometry optimization on an experimental quantum processor, estimating the equilibrium bond length of the dihydrogen molecule to within 0.014 Angstrom of the full configuration interaction value. Within the same experiment, we estimate the polarizability of the H2 molecule, finding agreement at the equilibrium bond length to within 0.06 a.u. (2% relative error).