Entangling gate performance and fidelity limits with neutral atom F\"orster resonances
Neutral-atom entangling gates are commonly analyzed with a single effective Rydberg-pair state, but near F\"orster resonances the pair manifold contains resonantly coupled interaction channels that change both the control landscape and the achievable fidelity. We develop a two-eigenstate model for this regime and show that when allowing for coupling to both pair states in the resonance, the gate fidelity is bounded by $\mathcal{F}\leq 1-(\pi/2)/(V\tau_R)$, for interaction strength $V$ and Rydberg lifetime $\tau_R$. We construct a gate protocol that saturates this bound in the large-Rabi-frequency limit, improving the existing fidelity limit by approximately $40\%$. We also evaluate common gate protocols near F\"orster resonances and find that retaining the exchange dynamics increases predicted fidelities by up to two orders of magnitude over earlier treatments.