  This work studies the variational quantum eigensolver algorithm, designed to
determine the ground state of a quantum mechanical system by combining
classical and quantum hardware. Methods of reducing the number of required
qubit manipulations, prone to induce errors, for the variational quantum
eigensolver are studied. We formally justify the qubit removal process as
sketched by Bravyi, Gambetta, Mezzacapo and Temme [arXiv:1701.08213 (2017)].
Furthermore, different classical optimization and entangling methods, both gate
based and native, are surveyed by computing ground state energies of H$_2$ and
LiH. This paper aims to provide performance-based recommendations for
entangling methods and classical optimization methods. Analyzing the VQE
problem is complex, where the optimization algorithm, the method of entangling,
and the dimensionality of the search space all interact. In specific cases
however, concrete results can be shown, and an entangling method or
optimization algorithm can be recommended over others. In particular we find
that for high dimensionality (many qubits and/or entanglement depth) certain
classical optimization algorithms outperform others in terms of energy error.
