VQE algorithm for chemistry simulation

The variational quantum eigensolver (VQE) [1] is a method that uses a hy- brid quantum-classical computational approach to find the eigenvalues of a Hamiltonian. VQE has been proposed as an alternative to fully quantum algorithms, which require hardware not yet available and has been success- fully applied to solve the electronic Schro ̈dinger equation for a variety of small molecules. The scalability of this method is limited by two factors: the complexity of the quantum circuits and the complexity of the classical optimization problem. Both of these factors are affected by the choice of the variational ansatz used to represent the trial wave function. The purpose of this work is to discuss the outlines of the VQE method, within the general framework of quantum computing. Therefore, I shall firstly present the core principles of quantum computing to solve compu- tational chemistry problems [2]. I shall then discuss briefly which limiting factors exist for our present computational technology and comment on the circuit models that can be used to describe both classic and quantum algo- rithms.[3, 4, 5] Finally I will focus on the rules that govern the action of these circuits, after introducing basic quantum chemistry concepts like the density operator and its matrix representation.[6] An historical view of the implementation of quantum computing algorithms will be also provided, to- gether with a comment on the potential usefulness of quantum computing to solve chemical computaitional problems.[7] Lastly some perspectives on the state of the art in quantum computing will be provided.[8, 9, 10, 11, 12]