A Tunable Multi-Qubit Coupling Element for Superconducting Circuits

We propose a SQUID tunable microwave coplanar waveguide (CPW) resonator to

implement programmable (up to all-to-all), tunable coupling between superconducting qubits. In

order to make qualitative and quantitative predictions about the behavior of this circuit element

as a tunable coupling element we develop a theoretical description of its dynamics. We will

outline a framework for analyzing dynamics in the presence of parametric flux modulation

through its boundary SQUIDs. This nonlinearity at the resonator boundary introduces mixing

between the central resonance of the system and the sidebands at integer multiples of the

modulation frequency. These sidebands yield an effective amplitude modulation of the resonance

mode profile from which we compute the resulting time dependent coupling of the tunable

resonator to the superconducting qubits. We present analytical and numerical results for the

coupling, demonstrating quantitative agreement. This time-dependent qubit-resonator coupling

can be used as the basis for two qubit gates and interactions in digital and analog quantum

computing protocols.