<p>Topology and symmetry play critical roles in characterizing quantum phases of matter. Recent advancements have unveiled symmetry-protected topological (SPT) phases in many-body systems as a unique class of short-range entangled states, notable for their nontrivial edge modes and characteristic ground-state entanglement gap. In this study, we demonstrate the robust simulation of many-body ground states of an Ising-cluster model on a quantum computer. By employing the method of quantum imaginary-time evolution (QITE) combined with enhanced zero-noise extrapolation techniques, we achieve accurate measurements of the transition between trivial and cluster SPT phases. Furthermore, we measured the characteristic edge modes and their associated topological entanglement properties, such as the second Rényi entropy, reduced density matrix, and entanglement spectral gap. Our work demonstrates the potential of using QITE in investigating sophisticated quantum phase transitions and critical phenomena on quantum computers.</p>