  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 {\em 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 quantum
imaginary time evolution (QITE) method. The effectiveness of SSQITE is
demonstrated through calculations of low-lying excited states of benchmark
model systems, including $\text{H}_2$ and $\text{LiH}$ molecules. With its
robustness in avoiding local minima, SSQITE shows promise for advancing quantum
computations of excited states across a wide range of applications.
