Quantum circuit implementation of the first-order perturbation in transverse-field Ising systems

The transverse-field Ising (TI) model is the standard prototype from which the properties of magnetic systems are often scrutinized. In addition, inhomogeneities in these systems can be treated as perturbations to the TI model. Incorporating perturbation escalates the difficulty of simulating such magnetic systems, specially if it is accompanied by an increase in system size or when higher dimensions or more complex lattice structures are considered.