  The variational quantum eigensolver (or VQE) uses the variational principle
to compute the ground state energy of a Hamiltonian, a problem that is central
to quantum chemistry and condensed matter physics. Conventional computing
methods are constrained in their accuracy due to the computational limits. The
VQE may be used to model complex wavefunctions in polynomial time, making it
one of the most promising near-term applications for quantum computing. Finding
a path to navigate the relevant literature has rapidly become an overwhelming
task, with many methods promising to improve different parts of the algorithm.
Despite strong theoretical underpinnings suggesting excellent scaling of
individual VQE components, studies have pointed out that their various
pre-factors could be too large to reach a quantum computing advantage over
conventional methods.
  This review aims to provide an overview of the progress that has been made on
the different parts of the algorithm. All the different components of the
algorithm are reviewed in detail including representation of Hamiltonians and
wavefunctions on a quantum computer, the optimization process, the
post-processing mitigation of errors, and best practices are suggested. We
identify four main areas of future research:(1) optimal measurement schemes for
reduction of circuit repetitions; (2) large scale parallelization across many
quantum computers;(3) ways to overcome the potential appearance of vanishing
gradients in the optimization process, and how the number of iterations
required for the optimization scales with system size; (4) the extent to which
VQE suffers for quantum noise, and whether this noise can be mitigated. The
answers to these open research questions will determine the routes for the VQE
to achieve quantum advantage as the quantum computing hardware scales up and as
the noise levels are reduced.
