  The calculation time for the energy of atoms and molecules scales
exponentially with system size on a classical computer but polynomially using
quantum algorithms. We demonstrate that such algorithms can be applied to
problems of chemical interest using modest numbers of quantum bits.
Calculations of the water and lithium hydride molecular ground-state energies
have been carried out on a quantum computer simulator using a recursive
phase-estimation algorithm. The recursive algorithm reduces the number of
quantum bits required for the readout register from about 20 to 4. Mappings of
the molecular wave function to the quantum bits are described. An adiabatic
method for the preparation of a good approximate ground-state wave function is
described and demonstrated for a stretched hydrogen molecule. The number of
quantum bits required scales linearly with the number of basis functions, and
the number of gates required grows polynomially with the number of quantum
bits.
