Investigation of strongly interacting quantum field theories (QFTs) remains one of the outstanding challenges of modern physics. Quantum simulation has the potential to be a crucial technique towards solving this problem. In this talk, I will describe analog quantum simulators for strongly interacting QFTs using mesoscopic quantum electronic circuit lattices. The tunable, robust and dispersive Josephson nonlinearity gives rise to the nonlinear interactions in these QFTs. I will concentrate on the quantum sine-Gordon model and its nontrivial generalizations in two space-time dimensions. In particular, I will show that superconducting quantum circuits provide a robust platform for probing exotic phenomenon like pure-quantum integrability occurring in the multi-field QFTs like the double sine-Gordon model, which do not admit a semi-classical description. I will describe the thermodynamic properties of the latter model using Bethe ansatz. Finally, I will show that the quantum-circuit-based lattice models of the aforementioned QFTs are amenable to numerical analysis using the density matrix renormalization group technique, which opens a new window into the rich entanglement properties of strongly interacting QFTs. In particular, I will describe various entanglement characteristics of the aforementioned QFTs.