  The idea to use quantum mechanical devices to simulate other quantum systems
is commonly ascribed to Feynman. Since the original suggestion, concrete
proposals have appeared for simulating molecular and materials chemistry
through quantum computation, as a potential ``killer application''. Indications
of potential exponential quantum advantage in artificial tasks have increased
interest in this application, thus, it is critical to understand the basis for
potential exponential quantum advantage in quantum chemistry. Here we gather
the evidence for this case in the most common task in quantum chemistry,
namely, ground-state energy estimation. We conclude that evidence for such an
exponential advantage across chemical space has yet to be found. While quantum
computers may still prove useful for quantum chemistry, it may be prudent to
assume exponential speedups are not generically available for this problem.
