The structured matrix completion problem (SMCP) is ubiquitous in several signal processing applications. In this article, we consider a fixed pattern, namely, the Hankel-structure for the SMCP under quantum formalism. By exploiting its structure, a lower-gate-complexity quantum circuit realization of a Hankel system is demonstrated. Further, we propose a quantum simulation algorithm for the Hankel-structured Hamiltonian with an advantage in quantum gate-operation complexity in comparison with the standard quantum Hamiltonian simulation technique. We show its application in eigenvalue spectrum estimation for signal processing applications. An error bound associated with this proposed quantum evolution is proposed with the consideration of spectrum estimation and measurement uncertainty. Numerical results are reported adopting random matrix theory in its fold to evaluate the efficacy of the proposed architecture and algorithm for large-dimensional systems, including an example application in delay estimation for ranging operations in a wireless communication system.