  Variational quantum algorithms exploit the features of superposition and
entanglement to optimize a cost function efficiently by manipulating the
quantum states. They are suitable for noisy intermediate-scale quantum (NISQ)
computers that recently became accessible to the worldwide research community.
Here, we implement and demonstrate the numerical processes on the 5-qubit and
7-qubit quantum processors on the IBM Qiskit Runtime platform. We combine the
commercial finite-element-method (FEM) software ABAQUS with the implementation
of Variational Quantum Eigensolver (VQE) to establish an integrated pipeline.
Three examples are used to investigate the performance: a hexagonal truss, a
Timoshenko beam, and a plane-strain continuum. We conduct parametric studies on
the convergence of fundamental natural frequency estimation using this hybrid
quantum-classical approach. Our findings can be extended to problems with many
more degrees of freedom when quantum computers with hundreds of qubits become
available in the near future.
