  As quantum computing technology improves and quantum computers with a small
but non-trivial number of N > 100 qubits appear feasible in the near future the
question of possible applications of small quantum computers gains importance.
One frequently mentioned application is Feynman's original proposal of
simulating quantum systems, and in particular the electronic structure of
molecules and materials. In this paper, we analyze the computational
requirements for one of the standard algorithms to perform quantum chemistry on
a quantum computer. We focus on the quantum resources required to find the
ground state of a molecule twice as large as what current classical computers
can solve exactly. We find that while such a problem requires about a ten-fold
increase in the number of qubits over current technology, the required increase
in the number of gates that can be coherently executed is many orders of
magnitude larger. This suggests that for quantum computation to become useful
for quantum chemistry problems, drastic algorithmic improvements will be
needed.
