  We introduce the framework of model space into quantum imaginary time
evolution (QITE) to enable stable estimation of ground and excited states using
a quantum computer. Model-space QITE (MSQITE) propagates a model space to the
exact one by retaining its orthogonality, and hence is able to describe
multiple states simultaneously. The quantum Lanczos (QLanczos) algorithm is
extended to MSQITE to accelerate the convergence. The present scheme is found
to outperform both the standard QLanczos and the recently proposed
folded-spectrum QITE in simulating excited states. Moreover, we demonstrate
that spin contamination can be effectively removed by shifting the imaginary
time propagator, and thus excited states with a particular spin quantum number
are efficiently captured without falling into the different spin states that
have lower energies. We also investigate how different levels of the unitary
approximation employed in MSQITE can affect the results. The effectiveness of
the algorithm over QITE is demonstrated by noise simulations for the H4 model
system.
