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The variational quantum eigensolver (VQE) and its variants, which is a methodfor finding eigenstates and eigenenergies of a given Hamiltonian, are appealingapplications of near-term quantum computers. Although the eigenenergies arecertainly important quantities which determines properties of a given system,their derivatives with respect to parameters of the system, such as positionsof nuclei if we target a quantum chemistry problem, are also crucial to analyzethe system. Here, we describe methods to evaluate analytical derivatives of theeigenenergy of a given Hamiltonian, including the excited state energy as wellas the ground state energy, with respect to the system parameters in theframework of the VQE. We give explicit, low-depth quantum circuits which canmeasure essential quantities to evaluate energy derivatives, incorporating withproof-of-principle numerical simulations. This work extends the theory of thevariational quantum eigensolver, by enabling it to measure more physicalproperties of a quantum system than before and to explore chemical reactions.

Theory of analytical energy derivatives for the variational quantum eigensolver