  Circuit quantum electrodynamics is one of the most promising platforms for
efficient quantum simulation and computation. In recent groundbreaking
experiments, the immense flexibility of superconducting microwave resonators
was utilized to realize hyperbolic lattices that emulate quantum physics in
negatively curved space. Here we investigate experimentally feasible settings
in which a few superconducting qubits are coupled to a bath of photons evolving
on the hyperbolic lattice. We compare our numerical results for finite lattices
with analytical results for continuous hyperbolic space on the Poincar\'{e}
disk. We find good agreement between the two descriptions in the
long-wavelength regime. We show that photon-qubit bound states have a
curvature-limited size. We propose to use a qubit as a local probe of the
hyperbolic bath, for example by measuring the relaxation dynamics of the qubit.
We find that, although the boundary effects strongly impact the photonic
density of states, the spectral density is well described by the continuum
theory. We show that interactions between qubits are mediated by photons
propagating along geodesics. We demonstrate that the photonic bath can give
rise to geometrically-frustrated hyperbolic quantum spin models with
finite-range or exponentially-decaying interaction.
