Josephson junctions. Similarly coupling of the qubits to their drives and Purcell relaxation rates of the qubit modes are related to the entry of the multiport impedance matrix connecting the qubits and the drive ports. This gives a complete microwave description of the system in the qubit subspace. The formulas requiring only evaluation at qubit frequencies(no need for frequency sweeps and fitting) make modeling and simulation of the chips much more efficient.

Simple relations of the qubit exchange coupling rates and the couplings of the qubits to the voltage drives to the impedance response allow application of microwave engineering techniques to improve the performance of the two-qubit gates. One application could be to use microwave coupler or filtering structures to shape the response profile to reduce unwanted terms in two-qubit gates.

Acknowledgements
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We thank Easwar Magesan and Hanhee Paik for useful discussions and Salvatore Olivadese for support with microwave simulations. DD acknowledges support from Intelligence Advanced Research Projects Activity (IARPA) under contract W911NF-16-0114.

Appendix
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\[sub:Derivation-of-the-Hamiltonian\]Derivation of the Hamiltonian for the Canonical Multiport Cauer Circuit
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Any multiport lossless impedance response can be synthesized with the canonical Cauer circuit shown in Fig. . The Cauer circuit consists of $N$ 