Interactions govern the flow of information and the formation of correlations in quantum systems, dictating the phases of matter found in nature and the forms of entanglement generated in the laboratory. Typical interactions decay with distance and thus produce a network of connectivity governed by geometry, e.g., by the crystalline structure of a material or the trapping sites of atoms in a quantum simulator. However, many envisioned applications in quantum simulation and computation require richer coupling graphs including nonlocal interactions, which notably feature in mappings of hard optimization problems onto frustrated spin systems and in models of information scrambling in black holes. Here, we report on the realization of programmable nonlocal interactions in an array of atomic ensembles within an optical cavity, where photons carry information between distant atomic spins. By programming the distance-dependence of interactions, we access effective geometries where the dimensionality, topology, and metric are entirely distinct from the physical arrangement of atoms. As examples, we engineer an antiferromagnetic triangular ladder, a Moebius strip with sign-changing interactions, and a treelike geometry inspired by concepts of quantum gravity. The tree graph constitutes a toy model of holographic duality, where the quantum system may be viewed as lying on the boundary of a higher-dimensional geometry that emerges from measured spin correlations. Our work opens broader prospects for simulating frustrated magnets and topological phases, investigating quantum optimization algorithms, and engineering new entangled resource states for sensing and computation.
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