  We present quantum algorithms, for Hamiltonians of linear combinations of
local unitary operators, for Hamiltonian matrix-vector products and for
preconditioning with the inverse of shifted reduced Hamiltonian operator that
contributes to the diagonal matrix elements only. The algorithms implement a
convergent series of approximations towards the exact solution of the full CI
(configuration interaction) problem. The algorithm scales with O(m^5 ), with m
the number of one-electron orbitals in the case of molecular electronic
structure calculations. Full CI results can be obtained with a scaling of
O(nm^5 ), with n the number of electrons and a prefactor on the order of 10 to
20. With low orders of Hamiltonian matrix-vector products, a whole repertoire
of approximations widely used in modern electronic structure theory, including
various orders of perturbation theory and/or truncated CI at different orders
of excitations can be implemented for quantum computing for both routine and
benchmark results at chemical accuracy. The lowest order matrix-vector product
with preconditioning, basically the second-order perturbation theory, is
expected to be a leading algorithm for demonstrating quantum supremacy for Ab
Initio simulations, one of the most anticipated real world applications. The
algorithm is also applicable for the hybrid variational quantum eigensolver.
