of the distributed linear passive structure the qubits are connected to). Such an approximate description in the harmonic basis is valid for qubits with low anharmonicity $\delta_{i}\ll\omega_{i}$ such as transmons. Qubit modes $i$ and $j$ are coupled to each other at exchange coupling rate $J_{ij}$ and the only remaining interaction between the qubit and resonator modes are the dispersive energy shifts $\chi_{ik}$’s.

We show that the exchange coupling rate $J_{ij}$ between qubit modes $i$ and $j$ in such an effective description is a simple function of the impedance response defined between the

$$J_{ij}=-\frac{1}{4}\sqrt{\frac{\omega_{i}\omega_{j}}{L_{i}L_{j}}}\mathrm{Im}\left[\frac{Z_{ij}\left(\omega_{i}\right)}{\omega_{i}}+\frac{Z_{ij}\left(\omega_{j}\right)}{\omega_{j}}\right]\label{eq:The-impedance-formula-for-J}$$

where $\omega_{i}$ is the frequency of the qubit $i$ given by $\omega_{i}=\omega_{J_{i}}-\frac{E_{C}^{(i)}/\hbar}{1-E_{C}^{(i)}/(\hbar\omega_{J_{i}})}$ with $\omega_{J_{i}}=1/\sqrt{L_{J_{i}}C_{i}}$ and $E_{C}^{(i)}=\frac{e^{2}}{2C_{i}}$ being the charging energy of the qubit $i$ of total shunt capacitance $C_{i}$. $L_{i}$ and $L_{j}$ are the “qubit inductances” corresponding to the qubits $i$ and $j$, respectively; related to the bare junction inductances $L_{J_{i}}$’s by $L_{i}=L_{J_{i}}/(1-\frac{2E_{C}^{(i)}}{\hbar\omega_{i}}$) such that $\omega_{i}=1/\sqrt{L_{i}C_{i}}$. $Z_{ij}\left(\omega\right)$ is the $\left(i,j\right)$-entry of the multiport impedance matrix

$$\varepsilon_{id}=\sqrt{\frac{\omega_{i}}{2\hbar L_{i}}}\mathrm{Im}\left[Z_{i,p(d)}(\omega_{i})\right]\frac{e^{i\theta_{d}}C_{p(d)}}{\sqrt{1+\omega_{d}^{2}Z_{0}^{2}C_{p(d)}^{2}}}\label{eq:epsilon-matrix}$$

where $\theta_{d}=\frac{\pi}{2}-\arctan(\omega_{d}Z_{0}C_{p(d)})$ and $Z_{i,p(d)}(\omega_{i})$ is the entry of the multiport impedance matrix connecting the drive port(with port index $p(d)$) corresponding to the voltage source $V_{d}$ ( to the qubit port $i$ evaluated at the frequency $\omega_{i}$ of qubit $i$; $\omega_{d}$ 