Hyperbolic Lattices in Circuit QED*

After two decades of development, superconducting circuits have emerged as a rich platform for quantum computation and simulation. Lattices of coplanar waveguide (CPW) resonators have been shown to produce artificial materials for microwave pho- tons, where interactions can be introduced either via non-linear resonator materials or qubit-resonator coupling. Here, we highlight the previously-overlooked property that these lattice sites are deformable and permit tight-binding lattices which are unattain- able, even in solid-state systems. Networks of CPW resonators can create a new class of materials, including lattices in an effective hyperbolic space with constant nega- tive curvature. We will discuss the mathematical tools needed to study the spectra of such structures and present experimental data showing that hyperbolic analogs of the kagome lattice in which photons in this lattice propagate along geodesics of the hy- perbolic metric, rather than along the standard straight lines of flat Euclidean, space can be produced in the lab. Additionally, we will highlight the prospects for stud- ies of information and entanglement propagation in hyperbolic geometries, as well as realization of other non-flat structures, such as trees or models with varying curvature.