  Recent advances in quantum computing devices have brought attention to hybrid
quantum-classical algorithms like the Variational Quantum Eigensolver (VQE) as
a potential route to practical quantum advantage in chemistry. However, it is
not yet clear whether such algorithms, even in the absence of device error,
could actually achieve quantum advantage for systems of practical interest. We
have performed an exhaustive analysis to estimate the number of qubits and
number of measurements required to compute the combustion energies of small
organic molecules and related systems to within chemical accuracy of
experimental values using VQE. We consider several key modern improvements to
VQE, including low-rank factorizations of the Hamiltonian. Our results indicate
that although these techniques are useful, they will not be sufficient to
achieve practical quantum computational advantage for our molecular set, or for
similar molecules. This suggests that novel approaches to operator estimation
leveraging quantum coherence, such as Enhanced Likelihood Functions
[arxiv:2006.09350, arxiv:2006.09349], may be required.
