A variety of photon-mediated operations are critical to the realization of scalable quantum information processing platforms and their accurate characterization is essential for the identification of optimal regimes and their experimental realizations. Such light-matter interactions are often studied with a broad variety of analytical and computational methods that are constrained by approximation techniques or by computational scaling. Quantum processors present a new avenue to address these challenges. Here, we develop and implement an algorithm compatible with NISQ (Noisy Intermediate Scale Quantum) devices that allows a comprehensive characterization of cavity mediated two-qubit gates over a broad parameter space. In particular, we investigate the underlying quantum state transfer between the qubits, and illustrate its efficiency as a function of the detunings and couplings between the qubits and the cavity, and as a function of the damping factor of the cavity. Our algorithm is validated by a comparison with analytical results where applicable. The algorithm allows us to identify under-explored regimes of optimal performance, from which we report two insightful findings- one where we observe a state transfer fidelity of about 99% in the intermediate detuning regime, and the other where we observe a significant boost in fidelity with qubits that are moderately detuned from each other by coupling them unequally to a far-detuned cavity. Furthermore, we illustrate the breakdown of the Rotating Wave Approximation (RWA) as we approach the ultra-strong coupling and the deep-dispersive regimes. While the current paper addresses a system that only involves a few qubits and can be solved using classical numerical methods, the algorithm can be readily extended to study a variety of systems and processes in quantum information processing that employ cavity or photon mediated coupling between qubits, particularly larger systems such as many-emitter ensembles, cavity arrays, and multimode cavities, that are classically intractable.