holds for typical parameter values.

\[sec:Anharmonicities-Chi-shifts\]Expressions for the Qubit Anharmonicities and the Dispersive Shifts in the Resonator Frequencies
==================================================================================================================================

In this section we derive expressions for the anharmonicity $\delta_{i}$ of the qubit mode $i$ and dispersive shift $\chi_{ik}$ in the frequency $\omega_{R_{k}}$ of the resonator mode $k$ due to qubit mode $i$ using the results of Appendix . Anharmonicities and dispersive shifts are generated by the nonlinear terms in the expansion of the junction potentials.

From the term $H_{\beta}$ in Eq. in the expansion in Eq. originally given in [@BBQ-Yale] we note the following

$$\begin{aligned}
\delta_{i} & = & -12\beta_{iiii}\\
\chi_{ik} & = & -24\beta_{iikk}\end{aligned}$$

Using the expression for $\beta_{pp'qq'}$ in Eq. and Eqs. and we obtain

$$\begin{aligned}
\delta_{i} & = & -E_{C}^{(i)}\left(\frac{\omega_{J_{i}}}{\omega_{i}}\right)^{2}\label{eq:lambda-anharmonicity}\\
\chi_{ik} & = & -2E_{C}^{(i)}\left(\frac{\omega_{J_{i}}^{2}}{\omega_{i}\omega_{R_{k}}}\right)r_{ki}^{2}C_{i}\left(\frac{\omega_{R_{k}}^{2}}{\omega_{R_{k}}^{2}-\omega_{i}^{2}}\right)^{2}\end{aligned}$$

From Eq. we note

$$r_{ik}=\frac{2g_{ik}}{\sqrt{C_{i}\omega_{i}\omega_{R_{k}}}}$$

Hence

$$\begin{aligned}
\chi_{ik} & = & -2E_{C}^{(i)}\left(\frac{\omega_{J_{i}}^{2}}{\omega_{i}\omega_{R_{k}}}\right)\left(\frac{4g_{ik}^{2}}{\omega_{i}\omega_{R_{k}}}\right)\left(\frac{\omega_{R_{k}}^{2}}{\omega_{R_{k}}^{2}-\omega_{i}^{2}}\right)^{2}\nonumber \\
 & = & 8\delta_{i}\left(\frac{g_{ik}\omega_{R_{k}}}{\omega_{R_{k}}^{2}-\omega_{i}^{2}}\right)^{2}\label{eq:chi-shift}\end{aligned}$$

Conclusion & Outlook
====================

We have analyzed superconducting quantum processors consisting of low anharmonicity transmon qubits. We have shown that the exchange coupling rates between qubits is related in a simple way to the off-diagonal entry of the multiport impedance matrix connecting the qubit ports evaluated at qubit frequencies. Qubit ports are defined across the 