Abstract
The variational quantum eigensolver (VQE) and its variants, which is a method for finding eigenstates and eigenenergies of a given Hamiltonian, are appealing applications of near-term quantum computers. Although the eigenenergies are certainly important quantities which determines properties of a given system, their derivatives with respect to parameters of the system, such as positions of nuclei if we target a quantum chemistry problem, are also crucial to analyze the system. Here, we describe methods to evaluate analytical derivatives of the eigenenergy of a given Hamiltonian, including the excited state energy as well as the ground state energy, with respect to the system parameters in the framework of the VQE. We give explicit, low-depth quantum circuits which can measure essential quantities to evaluate energy derivatives, incorporating with proof-of-principle numerical simulations. This work extends the theory of the variational quantum eigensolver, by enabling it to measure more physical properties of a quantum system than before and to explore chemical reactions.
Highlights
The variational quantum eigensolver (VQE) has attracted much attention as a potential application of near-term quantum computers [1]
This work extends the theory of the variational quantum eigensolver, by enabling it to measure more physical properties of a quantum system than before and to explore chemical reactions
We present a method to extract the derivatives of excited state energy based on the technique presented in Refs. [3,4]
Summary
The variational quantum eigensolver (VQE) has attracted much attention as a potential application of near-term quantum computers [1]. A simple way to compute energy derivatives is to use the finite difference method and calculate them numerically This approach, suffers from high computational costs as well as numerical errors and instabilities [7,9]. The number of energy points needed to evaluate the forces increases linearly to the number of atoms This high computational cost makes the numerical approach impractical in many cases. The analysis on the computational cost shows that the analytical differentiation provided in this work can be more practical than a finite difference approach, in a sense that one does not need to consider the best step size to extract the energy derivative information with an optimal precision. The presented methods extend the applicability of the VQE by enabling it to evaluate more physical properties than before
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