Circuit design implementing longitudinal coupling: a scalable scheme for superconducting qubits

We present a circuit construction for a new fixed-frequency superconducting qubit and show how it can be scaled up to a grid with strictly local interactions. The circuit QED realization we propose implements $\sigma_z$-type coupling between a superconducting qubit and any number of $LC$ resonators. The resulting \textit{longitudinal coupling} is inherently different from the usual $\sigma_x$-type \textit{transverse coupling}, which is the one that has been most commonly used for superconducting qubits. In a grid of fixed-frequency qubits and resonators with a particular pattern of always-on interactions, coupling is strictly confined to nearest and next-nearest neighbor resonators\footnote{P.-M. Billangeon et al., \textbf{Phy. Rev. B} 91:094517, 2015}; we note that just four distinct resonator frequencies, and only a single unique qubit frequency, suffice for the scalability of this scheme. There is never any direct coupling between the qubits. A controlled phase gate between two neighboring qubits can be realized with microwave drives on the qubits, without affecting the other qubits. This fact is a supreme advantage for the scalability of this scheme.