Engineering quantum spin liquids and many-body Majorana states with a driven superconducting box circuit

13 pages, including Figures and Tables. Version to appear in Physical Review BInternational audienceWe design a driven superconducting box with four spins S=1/2 (qubits) such that coupled devices can give insight on the occurrence of quantum spin liquids and many-body Majorana states. Within one box or island, we introduce a generalized nuclear magnetic resonance algorithm to realize our models and study numerically the spin observables in time as well as the emergent gauge fields. We discuss the stability of the box towards various detuning effects and we include dissipation effects through a Lindblad master equation. Coupling boxes allows us to realize quantum spin liquid phases of Kitaev ${\cal Z}_2$ spin models in various geometries with applications in the toric code. Quantum phase transitions and Majorana physics might be detected by measuring local susceptibilities. We show how to produce a N\' eel state of fluxes by coupling boxes and we address the role of local impurity fluxes leading to random Ising models. We also present an implementation of the Sachdev-Ye-Kitaev Majorana model in coupled ladder systems