  The variational quantum eigensolver (VQE) is a method that uses a hybrid
quantum-classical computational approach to find eigenvalues and eigenvalues of
a Hamiltonian. VQE has been proposed as an alternative to fully quantum
algorithms such as quantum phase estimation because fully quantum algorithms
require quantum hardware that will not be accessible in the near future. VQE
has been successfully applied to solve the electronic Schr\"{o}dinger equation
for a variety of small molecules. However, 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 choice of the variational ansatz used to represent the trial wave
function. Hence, the construction of efficacious ansatz is an active area of
research. Put another way, modern quantum computers are not capable of
executing deep quantum circuits produced by using currently available ansatze
for problems that map onto more than several qubits. In this review, we present
recent developments in the field of designing effective ansatzes that fall into
two categories -- chemistry inspired and hardware efficient -- that produce
quantum circuits that are easier to run on modern hardware. We discuss the
shortfalls of ansatzes originally formulated for VQE simulations, how they are
addressed in more sophisticated methods, and the potential ways for further
improvements.
