  In order to answer the problem of Quantum Phase Estimation Algorithm been not
suitable for NISQ devices, and allows one to outperform classical computers,
Variational Quantum Algorithms (VQAs) were designed. Our subject of interest is
the so-called Variational Quantum Eigensolver (VQE) algorithm and was
originally designed to simulate electronic structures and to compute the ground
state of a given molecule.
  VQE is made of two main components : an ansatz and a classical optimizer. The
ansatz runs on the quantum device and aims to simulate the wavefunction, the
parameters of the ansatz will be optimized until the expectation value is
minimum.
  The very first ansatz that has originally been used is called UCCSD and it is
based on Coupled Cluster Theory. The main issue considering UCCSD is the large
amount of parameters to optimize and this leads us to the introduction of
Adaptive Derivative-Assembled Pseudo-Trotter ansatz VQE (ADAPT-VQE) which
determines a quasi-optimal ansatz with a minimal number of parameters.
  The key point of ADAPT-VQE is to grow the ansatz at every step, by adding
operators chosen from a pre-determined pool of operators one-at-a-time,
assuring that the maximal amount of correlation energy is recovered at each
step. There exists different kind of ADAPT-VQE depending on the starting pool
of operators as the fermionic-ADAPT, the qubit-ADAPT or even the qubit
excitation based (QEB).
  Our goal is to implement the different types of ADAPT-VQE mentioned before.
After a quick review of the theoretical background under all of these concepts,
we will implement each algorithm using quiskit. We will also compare all of
these algorithms on different criterions such as the number of parameters, the
accuracy or the number of CNOT gate used on H2 and LiH molecules. Then we will
have a small discussion about the results we obtained.
