  Computing excited-state properties of molecules and solids is considered one
of the most important near-term applications of quantum computers. While many
of the current excited-state quantum algorithms differ in circuit architecture,
specific exploitation of quantum advantage, or result quality, one common
feature is their rooting in the Schr\"odinger equation. However, through
contracting (or projecting) the eigenvalue equation, more efficient strategies
can be designed for near-term quantum devices. Here we demonstrate that when
combined with the Rayleigh-Ritz variational principle for mixed quantum states,
the ground-state contracted quantum eigensolver (CQE) can be generalized to
compute any number of quantum eigenstates simultaneously. We introduce two
excited-state (anti-Hermitian) CQEs that perform the excited-state calculation
while inheriting many of the remarkable features of the original ground-state
version of the algorithm, such as its scalability. To showcase our approach, we
study several model and chemical Hamiltonians and investigate the performance
of different implementations.
