Arrays of Rydberg atoms interacting via dipole-dipole interactions offer a powerful platform for probing quantum many-body physics. However, these intrinsic interactions also determine and constrain the models—and parameter regimes thereof—for quantum simulation. Here, we propose a systematic framework to engineer arbitrary desired long-range interactions in Rydberg-atom lattices, enabling the realization of fully tunable $J_1\text{−}{J}_2\text{−}{J}_3$ Heisenberg models. Using site-resolved periodic modulation of Rydberg states, we develop an experimentally feasible protocol to precisely control the interaction ratios $J_2/J_1$ and $J_3/J_1$ in a kagome lattice. This control can increase the effective range of interactions and drive transitions between competing spin-ordered and spin-liquid phases. To generalize this approach beyond the kagome lattice, we reformulate the design of modulation patterns through a graph-theoretic approach, demonstrating the universality of our method across all 11 planar Archimedean lattices. Our strategy overcomes the inherent constraints of power-law-decaying dipolar interactions, providing a versatile toolbox for exploring frustrated magnetism, emergent topological phases, and quantum correlations in systems with long-range interactions.

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• Quantum simulation
• Topological phases of matter
• Rydberg atoms & molecules
• Heisenberg model