  Quantum annealing (QA) is one of the ways to search the ground state of the
problem Hamiltonian. Here, we propose the QA scheme to search arbitrary excited
states of the problem Hamiltonian. In our scheme, an $n$-th excited state of
the trivial Hamiltonian is initially prepared and is adiabatically changed into
an $n$-th excited state of the target Hamiltonian. Although our scheme is
general such that we can search any excited states, we especially discuss the
first excited state search in this paper. As a comparison, we consider a
non-adiabatic scheme to find the first excited state with non-adiabatic
transitions from the ground state. By solving the Lindblad master equation, we
evaluate the performance of each scheme under the influence of decoherence. Our
conclusion is that the adiabatic scheme show better performance than the
non-adiabatic scheme as long as the coherence time of qubits is sufficiently
long. These results are important for applications in the area of quantum
chemistry, quantum simulation, and post-quantum cryptography.
