Galvanic coupling of flux qubits: simple theory and tunability

Galvanic coupling of small-area (three-junction) flux qubits, using shared large Josephson junctions, has been shown to yield appreciable interaction strengths in a flexible design, which does not compromise the junctions' intrinsic good coherence properties. For an introduction, I recapitulate an elementary derivation of the coupling strength, which is subsequently generalized to the case of tunable coupling for a current-biased shared junction. While the ability to vary coupling constants by, say, 20% would be useful in experiments, sign-tunability (implying switchability) is highly preferable for several quantum-computing paradigms. This note sketches two ideas: a "crossbar" design with competing ferro- and antiferromagnetic current-biased tunable couplings, and a "mediated" one involving an extra loop between the qubits. The latter is a variation on proposals for tunable capacitive coupling of charge qubits, and tunable inductive coupling of large-area flux qubits.