Non-Perturbative Geometric Framework for Single-Qubit Gates under Always-On Couplings

In qubit arrays with always-on couplings, single-qubit gates pose a control challenge often as demanding as entangling operations. The same interactions that enable two-qubit entanglement induce crosstalk that significantly degrades single-qubit fidelity. We present a non-perturbative analytical framework for constructing high-fidelity single-qubit gates in the presence of such couplings. From the geometric structure of SU(2) dynamics, we derive a crosstalk-suppression criterion. The dynamics must trace closed loops on a 2-sphere, with a net-zero enclosed-area condition arising when zero-detuning subspaces are present, and the pulse waveform corresponds to the geodesic curvature of the loop. Unlike previous Euclidean-geometric and perturbative dynamically-corrected-gate constructions, the framework operates on a 2-sphere whose intrinsic curvature is set by the detuning, enabling crosstalk suppression even when couplings are comparable to the drive amplitude. Noise robustness enters as an additional constraint along the same closed loop via the Magnus expansion. In two- and three-qubit Heisenberg chains, the resulting pulses are robust against fluctuations in both coupling strength and qubit frequency. Our pulses outperform a representative perturbative robust-control pulse by more than an order of magnitude in fidelity when the always-on coupling approaches the drive amplitude, where perturbative methods break down.