Analysis of a tuneable coupler for superconducting phase qubits

This paper presents a theoretical analysis of the recently realized tuneable coupler for superconducting phase qubits (R. C. Bialczak et al., Ref.
\protect\onlinecite{Bialczak}). The coupling can be turned off by compensating a negative mutual inductance with a tuneable Josephson inductance. The main coupling in this system is of the $XX$ type and can be zeroed exactly, while there is also a small undesired contribution of the $ZZ$ type. We calculate both couplings as functions of the tuning parameter (bias current) and focus on the residual coupling in the OFF regime. In particular, we show that for typical experimental parameters the coupling OFF/ON ratio is few times ${10}^{-3}$, and it may be zeroed by proper choice of parameters. The remaining errors due to physical presence of the coupler are on the order of ${10}^{-6}$.