  Quantum chemistry calculations are important applications of quantum
annealing. For practical applications in quantum chemistry, it is essential to
estimate a ground state energy of the Hamiltonian with chemical accuracy.
However, there are no known methods to guarantee the accuracy of the estimation
of the energy calculated by quantum annealing. Here, we propose a way to
improve the accuracy of the estimate of the ground state energy by combining
quantum annealing with classical computation. In our scheme, before running the
QA, we need a pre-estimation of the energies of the ground state and first
excited state with some error bars (corresponding to possible estimation error)
by performing classical computation with some approximations. We show that, if
an expectation value and variance of the energy of the state after the QA are
smaller than certain threshold values (that we can calculate from the
pre-estimation), the QA provides us with a better estimate of the ground state
energy than that of the pre-estimation. Since the expectation value and
variance of the energy can be experimentally measurable by the QA, our results
pave the way for accurate estimation of the ground state energy with the QA.
