  We show how quantum many-body systems on hyperbolic lattices with
nearest-neighbor hopping and local interactions can be mapped onto quantum
field theories in continuous negatively curved space. The underlying lattices
have recently been realized experimentally with superconducting resonators and
therefore allow for a table-top quantum simulation of quantum physics in curved
background. Our mapping provides a computational tool to determine observables
of the discrete system even for large lattices, where exact diagonalization
fails. As an application and proof of principle we quantitatively reproduce the
ground state energy, spectral gap, and correlation functions of the
noninteracting lattice system by means of analytic formulas on the Poincar\'{e}
disk, and show how conformal symmetry emerges for large lattices. This sets the
stage for studying interactions and disorder on hyperbolic graphs in the
future. Importantly, our analysis reveals that even relatively small discrete
hyperbolic lattices emulate the continuous geometry of negatively curved space,
and thus can be used to experimentally resolve fundamental open problems at the
interface of interacting many-body systems, quantum field theory in curved
space, and quantum gravity.
