“Purcell Loss”. In this section we compute rates for the Purcell loss of the qubit modes we identified in the earlier sections.

As in Section the coupling of the qubits to external electronics is modeled with the idealized circuit model in Fig. and we will use the same coupling matrices of the formalism in [@Burkard] that we calculated in Appendix for the drive couplings. We have $N_{D}$ baths corresponding to tranmission lines of characteristic impedances $Z_{0}$’s driving the qubits as shown in Fig. . Assuming couplings of qubits to the lines are small, to first order in these couplings, we will assume that $T_{1}$ rates can be computed separately for each bath. The total rate will then be the sum of rates due to each line.

We start by noting that when we have only the bath due to the drive line of the voltage source $V_{d}$ with port index $p(d)$ $\mathbf{C}_{D}$ defined in Eq. is a scalar $C_{p(d)}$ for $1\leq d\leq N_{D}$. Hence $\bar{\mathbf{m}}$ in Eq. is

$$\bar{\mathbf{m}}_{d}=-C_{p(d)}\left(\begin{array}{c}
\mathbf{0}_{N\times1}\\
\mathbf{v}_{d}
\end{array}\right)$$

where $\mathbf{v}_{d}=(\begin{array}{ccc}
v_{1d} & \ldots & v_{Md}\end{array})^{T}$ is the $d$-th column of the matrix $\mathbf{V}$ corresponding to the drive line with port index $p(d)$. After the Schrieffer-Wolff transformation 