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.