A scalable superconducting circuit framework for emulating physics in hyperbolic space

Recent theoretical and experimental advances have revealed distinct physical phenomena physical phenomena in hyperbolic space, absent in materials based on Euclidean geometries. We present a scalable superconducting circuit framework for the analogue quantum emulation of tight-binding models on hyperbolic and kagome-like lattices. Unlike previous implementations that encoded the hyperbolic metric through physical distances, our approach encodes it directly into the capacitive couplings between high-Q superconducting resonators. This design overcomes key limitations of earlier efforts by enabling genuine approximations of hyperbolic connectivity, higher spectral resolution, and scalability unconstrained by geometric contraction. Using this method, we experimentally realize three distinct lattices—including, for the first time to our knowledge, a hyperbolic lattice whose unit cell is embedded on a genus-3 Riemann surface. The measured spectra distinguish the hyperbolic lattice from its kagome-like counterpart through the absence of a degenerate flat band. Our results establish a versatile platform for large-scale studies of hyperbolic materials and lay the groundwork for exploring holographic phenomena, higher-dimensional band structures, and hyperbolic quantum error correction codes within circuit-QED architectures.