Optimal Connectivity from Idle Qubit residual coupling Cross-Talks in a Cavity Mediated Entangling Gate

Quantum processors operated through long range interaction mediated by a microwave resonator have been envisioned to allow for high connectivity. The ability to selectively operate qubits rely on the possibility to dynamically suppress the coupling between each qubit and the resonator, however there always remains a residual coupling. In this article, we investigate the effect of high processor connectivity on average two qubit gate fidelity in a cavity based architecture with tunable coupling. Via a perturbative approach, we quantify the cross-talk errors from transverse residual couplings and show that they scale as $nm^2$ where $n$ is the number of idle qubits and $m$ is the ratio between the transverse residual and active couplings. Setting an error threshold $E_\mathrm{thr}$, we demonstrate that cross-talks restrict the hardware topology and prevent the full use of all-to-all connectivity. We predict that the maximum number of qubits allowed by $E_\mathrm{thr}$ scales as $n \propto E_\mathrm{thr}/m^2$.