is the frequency of the voltage source $V_{d}$(assuming a single tone sinusoidal signal), $Z_{0}$ is the characteristic impedance of the drive lines which is typically $Z_{0}=50\Omega$ and $C_{p(d)}$ is the shunting capacitance of the drive port corresponding to the voltage source $V_{d}$

$$\frac{1}{T_{1}^{i,d}}=\frac{2}{L_{i}}\mathrm{Im}\left[Z_{i,p(d)}\left(\omega_{i}\right)\right]^{2}\frac{\omega_{i}^{2}Z_{0}C_{p(d)}^{2}}{1+\omega_{i}^{2}Z_{0}^{2}C_{p(d)}^{2}}$$

We note here that all the dispersive rates of qubit-qubit interactions and of interactions of qubits with the external electronics are functionals of the the multiport impedance function $\mathbf{Z}(\omega)$ and bare junction inductances $L_{J_{i}}$’s since the shunting capacitances $C_{i}$’s of the qubit ports are related to the residue $\mathbf{A}_{0}$ of $\mathbf{Z}(\omega)$ at DC as given in Eq. (Same argument applies to the shunt capacitances $C_{p(d)}$’s of the drive ports) and the qubit frequencies $\omega_{i}$’s and anharmonicities $\delta_{i}$’s are functions of qubit shunt capacitances and bare junction inductances.

\[sec:Derivation-of-J-couplings\]Derivation of the Formula for the Exchange Coupling Rates between the Qubits
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Assuming we have $N$ Josephson junctions in the circuit we define the $N\times N$ multiport impedance matrix $\mathbf{Z}$ seen looking into qubit ports defined across the junction terminals ($\mathbf{Z}$ has to be evaluated without shunting the ). Neglecting all the losses we can write the following partial fraction expansion for the imaginary part of $\mathbf{Z}\left(\omega\right)$ as a 