  The variational quantum eigensolver (VQE), a variational algorithm to obtain
an approximated ground state of a given Hamiltonian, is an appealing
application of near-term quantum computers. The original work [A. Peruzzo et
al.; \textit{Nat. Commun.}; \textbf{5}, 4213 (2014)] focused only on finding a
ground state, whereas the excited states can also induce interesting phenomena
in molecules and materials. Calculating excited states is, in general, a more
difficult task than finding ground states for classical computers. To extend
the framework to excited states, we here propose an algorithm, the
subspace-search variational quantum eigensolver (SSVQE). This algorithm
searches a low energy subspace by supplying orthogonal input states to the
variational ansatz and relies on the unitarity of transformations to ensure the
orthogonality of output states. The $k$-th excited state is obtained as the
highest energy state in the low energy subspace. The proposed algorithm
consists only of two parameter optimization procedures and does not employ any
ancilla qubits. The disuse of the ancilla qubits is a great improvement from
the existing proposals for excited states, which have utilized the swap test,
making our proposal a truly near-term quantum algorithm. We further generalize
the SSVQE to obtain all excited states up to the $k$-th by only a single
optimization procedure. From numerical simulations, we verify the proposed
algorithms. This work greatly extends the applicable domain of the VQE to
excited states and their related properties like a transition amplitude without
sacrificing any feasibility of it.
