  After close to two decades of research and development, superconducting
circuits have emerged as a rich platform for both quantum computation and
quantum simulation. Lattices of superconducting coplanar waveguide (CPW)
resonators have been shown to produce artificial materials for microwave
photons, where weak interactions can be introduced either via non-linear
resonator materials or strong interactions via qubit-resonator coupling. Here,
we highlight the previously-overlooked property that these lattice sites are
deformable and allow the realization of tight-binding lattices which are
unattainable, even in conventional solid-state systems. In particular, we show
that networks of CPW resonators can create a new class of materials which
constitute regular lattices in an effective hyperbolic space with constant
negative curvature. We present numerical simulations of a series of hyperbolic
analogs of the kagome lattice which show unusual densities of states with a
spectrally-isolated degenerate flat band. We also present a proof-of-principle
experimental realization of one of these lattices. This paper represents the
first step towards on-chip quantum simulation of materials science and
interacting particles in curved space.
