  A quantum simulator of U(1) lattice gauge theories can be implemented with
superconducting circuits. This allows the investigation of confined and
deconfined phases in quantum link models, and of valence bond solid and spin
liquid phases in quantum dimer models. Fractionalized confining strings and the
real-time dynamics of quantum phase transitions are accessible as well. Here we
show how state-of-the-art superconducting technology allows us to simulate
these phenomena in relatively small circuit lattices. By exploiting the strong
non-linear couplings between quantized excitations emerging when
superconducting qubits are coupled, we show how to engineer gauge invariant
Hamiltonians, including ring-exchange and four-body Ising interactions. We
demonstrate that, despite decoherence and disorder effects, minimal circuit
instances allow us to investigate properties such as the dynamics of electric
flux strings, signaling confinement in gauge invariant field theories. The
experimental realization of these models in larger superconducting circuits
could address open questions beyond current computational capability.
