From Universal to Useful: Local Efficiency in Symmetry-constrained Universal Tunnel-coupled Quantum Simulators

Neutral-atom platforms, including tunnel-coupled optical tweezer arrays and optical superlattices, are emerging as versatile quantum hardware for simulating strongly correlated dynamics and potentially for broader computational tasks. This promise relies on "universality": given the hardware-native gate set, one can in principle approximate arbitrary state transformations. However, continuous symmetries—such as particle number conservation or spin symmetries—are ubiquitous, and with these constraints it is not obvious whether universality implies locally efficient state transformations, as it does when the symmetries are absent. Here, we ask a consequential question: can continuous symmetries force certain target unitaries, even those supported on a small subregion, to require inefficient, system-wide compilation when one is restricted to local symmetry-preserving gates? We develop a Lie-algebra-based theory framework that identifies which operations allow locally efficient synthesis within tunnel-coupled neutral-atom schemes, and which face intrinsic symmetry-induced obstructions. These results clarify the practical scope of programmability in tunnel-coupled platforms beyond analog quantum simulation.