Ultrastrong capacitive and inductive couplings of flux qubits*

We analyse interactions between two 3-Josephson junctions flux qubits with a general many-body projector into the low energy subspace. Specifically, we obtain the effective Hamiltonian of two qubits coupled via a capacitor and/or a Josephson junction. We show that those two elements allow engineering a fairly large family of qubit-qubit Hamiltonians with XX, YY and ZZ terms, including relevant models for the field of quantum computing as strongly coupled or non-stoquastic ones. We consider all the ways of connecting the two qubits and extract the best experimentally realizable configuration that produces a large interaction (compared with the qubit gap) while keeping a good qubit anharmonicity.