  We developed a quantum eigensolver (QE) which is based on an extension of
optimized binary configurations measured by quantum annealing (QA) on a D-Wave
Quantum Annealer (D-Wave QA). This approach performs iterative QA measurements
to optimize the eigenstates $\vert \psi \rangle$ without the derivation of a
classical computer. The computational cost is $\eta M L$ for full eigenvalues
$E$ and $\vert \psi \rangle$ of the Hamiltonian $\hat{H}$ of size $L \times L$,
where $M$ and $\eta$ are the number of QA measurements required to reach the
converged $\vert \psi \rangle$ and the total annealing time of many QA shots,
respectively. Unlike the exact diagonalized (ED) algorithm with $L^3$
iterations on a classical computer, the computation cost is not significantly
affected by $L$ and $M$ because $\eta$ represents a very short time within
$10^{-2}$ seconds on the D-Wave QA. We selected the tight-binding $\hat{H}$
that contains the exact $E$ values of all energy states in two systems with
metallic and insulating phases. We confirmed that the proposed QE algorithm
provides exact solutions within the errors of $5 \times 10^{-3}$. The QE
algorithm will not only show computational supremacy over the ED approach on a
classical computer but will also be widely used for various applications such
as material and drug design.
