The structural design of arch roofs or bridges requires the analysis of their unstable behaviors depending on certain parameters defined in the arch shape. Their maintenance should estimate the parameters from observed data. However, since the critical parameters exist in the equilibrium paths of the arch, and a small change in such the parameters causes a significant change in their behaviors. Thus, estimation to find the critical ones should be carried out using a global search algorithm. In this paper we study the parameter estimation for a shallow arch by a quantum-inspired evolution algorithm. A cost functional to estimate the system parameters included in the arch consists of the difference between the observed signal and the estimated signal of the arch system. The design variables are shape, external load and damping constant in the arch system. We provide theoretical and numerical examples for estimation of the parameters from both contaminated data and pure data.