Analytical controls for dispersive multi-qubit interactions in mediator-coupled quantum registers

Fast control of long-lived quantum registers often relies on an auxiliary degree of freedom, a mediator, but real mediator excitation can introduce decoherence, leakage, or loss when the mediator is not the coherence-optimized component. We introduce dispersive interaction via analytical linear inversion (DIAL), an analytical control construction for excitation-suppressed interaction engineering in mediator-coupled quantum registers. In the dispersive regime, we show that off-resonant driving of an effective two-level mediator transition yields a diagonal Hamiltonian whose Pauli-$Z$ interaction rates depend linearly on the drive intensities through a transfer matrix determined by the register-resolved transition spectrum. This establishes a direct linear map from experimentally accessible controls to induced multi-qubit interactions, enabling control design without iterative propagation of the full driven dynamics. We turn this structure into the target-aware DIAL algorithm, which selects favorable off-resonant tones and determines their drive intensities for a prescribed target interaction. We benchmark the resulting controls across 100 register realizations for each register size $n=2,3,4,5$, targeting the highest-order interaction $Z_1\cdots Z_n$ with an interaction phase of magnitude $\pi \mathrm{/}4$, which generates an $n$-qubit GHZ state up to local equivalence. Propagation under the full driven mediator-register dynamics yields typical gate fidelities above $0.997$, while the maximum transient mediator excitation remains near $1\mathrm{\%}$ on average, far below what is typically possible with resonant Rabi drives. The manuscript is accompanied by an open-source Python package implementing the target-aware DIAL algorithm and the associated numerical validation workflow.