Physical implementations of qubit coupling

Coupling by linear passive elements. Capacitive coupling

To have good basic elements is not enough – it is necessary to be able to connect them in a controllable way, without losing quantum coherence. Any simple effective coupling Hamiltonian (like in Eqs. (3.81, 3.82)) must be somehow implemented “in metal”. Here superconducting circuits provide a wide variety of coupling schemes to choose from (see, e.g., Wendin and Shumeiko, 2005). We will begin with the simplest case, when the interaction between the qubits is realized using linear elements (conventional capacitances and inductances), the coupling circuit stays in its ground state and adiabatically follows the evolution of the qubits (Averin and Bruder, 2003) – that is, it remains “passive”. For this to happen, the excitation energy of the coupler, ħωres, must be much higher than the interlevel spacing in the qubits (where ωres is the resonance frequency of the coupler). In other words, the evolution of the coupler is much faster than that of the qubits, and the coupler can indeed adjust to changes in the state of the latter. In the case of a purely capacitive or purely inductive coupling this condition is automatically satisfied, as then ωres → ∞.