  Quantum chemistry and materials is one of the most promising applications of
quantum computing. Yet much work is still to be done in matching
industry-relevant problems in these areas with quantum algorithms that can
solve them. Most previous efforts have carried out resource estimations for
quantum algorithms run on large-scale fault-tolerant architectures, which
include the quantum phase estimation algorithm. In contrast, few have assessed
the performance of near-term quantum algorithms, which include the variational
quantum eigensolver (VQE) algorithm. Recently, a large-scale benchmark study
[Gonthier et al. 2020] found evidence that the performance of the variational
quantum eigensolver for a set of industry-relevant molecules may be too
inefficient to be of practical use. This motivates the need for developing and
assessing methods that improve the efficiency of VQE. In this work, we predict
the runtime of the energy estimation subroutine of VQE when using robust
amplitude estimation (RAE) to estimate Pauli expectation values. Under
conservative assumptions, our resource estimation predicts that RAE can reduce
the runtime over the standard estimation method in VQE by one to two orders of
magnitude. Despite this improvement, we find that the runtimes are still too
large to be practical. These findings motivate two complementary efforts
towards quantum advantage: 1) the investigation of more efficient near-term
methods for ground state energy estimation and 2) the development of problem
instances that are of industrial value and classically challenging, but better
suited to quantum computation.
