Symplectic Schrieffer-Wolff perturbation theory for superconducting circuits with nonreciprocal devices

Superconducting circuits are one of the leading platforms for realizing large-scale quantum processors. In superconducting quantum processors with a large number of qubits the characterization of effective parameters modelling the circuit is a difficult task. In this work we obtain linear effective couplings between the qubits for superconducting processors formed by low-anharmonicity qubits dispersively coupled through general linear (non)reciprocal devices by applying a full-fledged symplectic version of the Schrieffer-Wolff transformation. We show that the analytical expressions of such couplings are connected to the immittance response matrices of the system. Our work extends the quantum engineer's toolbox to characterize and design processors of superconducting circuits.