Simulating molecules using the VQE algorithm on Qiskit

Abstract

Introduction Feynman’s ideas to employ quantum systems for simulating other quantum systems [1] gave rise to quantum simulation [2]. While current quantum computers are still prone to decoherence and rely on error correction [3], the development of hybrid algorithms that employ both quantum and classical computation allow to perform quantum simulation [4]. Among these algorithms, the Variational Quantum Eigensolver (VQE) algorithm has permitted to explore the electronic structure of simple atoms and molecules by exploiting the Rayleigh–Ritz variational principle [5]. This has enabled theoretical chemists to gain insight about reaction rates, find chemically stable structures, determine spectroscopic properties of molecules, among others. The VQE employs a parameterized circuit that prepares a quantum state that aims to approach the ground-state of the molecular Hamiltonian. The expectation value of this Hamiltonian, calculated by a procedure known as Hamiltonian averaging [6], provides a cost function to minimize by optimizing the set of parameters. This optimization procedure is performed using a classical computer. These steps are performed iteratively until convergence is found. Among some of the main challenges of VQE is it’s dependence on the variational form to find the global minimum for the energy expectation value. Both hardware efficient variational forms [7], that employ a limited set of gates, and chemically inspired [8] that adapt classical computational chemistry methods and take into account the chemical details, have been proposed. In this supplementary material, we introduce the technical application of VQE in quantum chemistry by calculating the ground-state single point energy of the H2 molecule. Furthermore, we explore the Energy potential surface over a range of distances and discuss the effect of the variables on the accuracy of the results. Together, we provide the implementation of the VQE algorithm for finding the ground-state energy of the H2 molecule on Qiskit [9] library for python.