The topological state of matter is a potential resource to realize long-term fault-tolerant quantum computers beyond the near-term noisy intermediate-scale quantum devices. To achieve the realization, we need a deep understanding of topological behaviors in real quantum computers. However, quantum-circuit algorithms to analyze topological properties have still been insufficient. Here we propose three quantum-circuit algorithms, (i) to find the ground state in the selected parity subspace, (ii) to determine the many-body topological invariant, and (iii) to visualize the zero-energy edge mode. To demonstrate these algorithms, we adopt the interacting Kitaev chain as a typical model of many-body topological superconductors in one dimension. The algorithms are applicable to not only one-dimensional topological superconductors but other topological states including higher-dimensional systems.