  The computational cost of exact methods for quantum simulation using
classical computers grows exponentially with system size. As a consequence,
these techniques can only be applied to small systems. By contrast, we
demonstrate that quantum computers could exactly simulate chemical reactions in
polynomial time. Our algorithm uses the split-operator approach and explicitly
simulates all electron-nuclear and inter-electronic interactions in quadratic
time. Surprisingly, this treatment is not only more accurate than the
Born-Oppenheimer approximation, but faster and more efficient as well, for all
reactions with more than about four atoms. This is the case even though the
entire electronic wavefunction is propagated on a grid with appropriately short
timesteps. Although the preparation and measurement of arbitrary states on a
quantum computer is inefficient, here we demonstrate how to prepare states of
chemical interest efficiently. We also show how to efficiently obtain
chemically relevant observables, such as state-to-state transition
probabilities and thermal reaction rates. Quantum computers using these
techniques could outperform current classical computers with one hundred
qubits.
