  Quantum annealing is a way to prepare an eigenstate of the problem
Hamiltonian. Starting from an eigenstate of a trivial Hamiltonian, we slowly
change the Hamiltonian to the problem Hamiltonian, and the system remains in
the eigenstate of the Hamiltonian as long as the so-called adiabatic condition
is satisfied. By using devices provided by D-Wave Systems Inc., there were
experimental demonstrations to prepare a ground state of the problem
Hamiltonian. However, up to date, there are no demonstrations to prepare the
excited state of the problem Hamiltonian with quantum annealing. Here, we
demonstrate the excited-state search by using the D-wave processor. The key
idea is to use the reverse quantum annealing with a hot start where the initial
state is the excited state of the trivial Hamiltonian. During the reverse
quantum annealing, we control not only the transverse field but also the
longitudinal field and slowly change the Hamiltonian to the problem Hamiltonian
so that we can obtain the desired excited state. As an example of the exited
state search, we adopt a two-qubit Ising model as the problem Hamiltonian and
succeed to prepare the excited state. Also, we solve the shortest vector
problem where the solution is embedded into the first excited state of the
Ising Hamiltonian. Our results pave the way for new applications of quantum
annealers to use the excited states.
