  We have analyzed and proposed coupling mechanisms between Three Josephson
Junction Flux Qubits (3JJQ). For this, we have developed a numerical method to
extract the effective Hamiltonian of a system of coupled qubits via the
Schrieffer-Wolff transformation (SWT). We then give a comprehensive
introduction to the 3JJQ, and study it analytically by approximating its
potential with a Harmonic well. With a clear understanding of the 3JJQs, we use
the SWT to gain intuition about their effective dipolar interaction with the
electromagnetic field, and use that intuition to propose and study analytically
and numerically the capacitive coupling of two 3JJQs via a non-tunable
capacitor, and the inductive coupling of two 3JJQs via a tunable Josephson
Junction (dc-SQUID), showing that we are able to reproduce non-stoquastic
Hamiltonians in the strong-coupling regime.
