Quantum
systems in excited states are attracting significant interest
with the advent of noisy intermediate-scale quantum (NISQ) devices.
While ground states of small molecular systems are typically explored
using hybrid variational algorithms like the variational quantum eigensolver
(VQE), the study of excited states has received much less attention,
partly due to the absence of efficient algorithms. In this work, we
introduce the subspace search quantum imaginary time evolution (SSQITE) method, which calculates excited states using quantum devices
by integrating key elements of the subspace search variational quantum
eigensolver (SSVQE) and the variational quantum imaginary time evolution
(VarQITE) method. The effectiveness of SSQITE is demonstrated through
calculations of low-lying excited states of benchmark model systems
including H2 and LiH molecules. A toy Hamiltonian is also
employed to demonstrate that the robustness of VarQITE in avoiding
local minima extends to its use in excited state algorithms. With
this robustness in avoiding local minima, SSQITE shows promise for
advancing quantum computations of excited states across a wide range
of applications.