Ultrastrong Capacitive Coupling of Flux Qubits

A flux qubit can interact strongly when it is capacitively coupled to other circuit elements. This interaction be separated into two parts, one acting on the subspaces and in which excited states mediate interaction. The first term dominates between an $LC$ resonator, leading ultrastrong couplings of form ${\ensuremath{\sigma}}^{y}(a+{a}^{\ifmmode\dagger\else\textdagger\fi{}})$, complement inductive ${\ensuremath{\sigma}}^{x}i({a}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{-}a)$ coupling. However, coupling qubits capacitively, all terms need taken account, complex nonstoquastic types ${\ensuremath{\sigma}}_{1}^{y}{\ensuremath{\sigma}}_{2}^{y}$, ${\ensuremath{\sigma}}_{1}^{z}{\ensuremath{\sigma}}_{2}^{z}$, ${\ensuremath{\sigma}}_{1}^{x}{\ensuremath{\sigma}}_{2}^{x}$. Our theory explains these interactions, describing them general properties---coupling capacitances, gaps, inductive, Josephson capacitive energies---that apply a wide variety circuits designs.