Recent advance in quantum simulations of interacting photons using superconducting circuits offers opportunities for investigating the Bose-Hubbard model in various geometries with hopping coefficients and self-interactions tuned to both signs. Here we investigate phenomena related to localized states associated with a flat-band supported by the saw-tooth geometry. A localization-delocalization transition emerges in the non-interacting regime as the sign of hopping coefficient is changed. In the presence of interactions, patterns of localized states approach a uniform density distribution for repulsive interactions while interesting localized density patterns can arise in strongly attractive regime. The density patterns indicate the underlying inhomogeneity of the simulator. Two-particle correlations can further distinguish the nature of the localized states in attractive and repulsive interaction regimes. We also survey possible experimental implementations of the simulator.
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