apply the impedance formula for $J$ coupling to the arrangement shown in Fig. where we have four qubits on each bus. We model each bus as a simple $LC$ resonator at $6.30\,GHz$ capacitively coupled to qubits. Using realistic parameter values corresponding to a real device fabricated at IBM we obtain the decay plots in Fig. which confirm exponential decay of $J$ couplings over the chips.

![\[fig:2by8-J-scaling-plot\]Exponential decay of the $J_{1k}$ coupling rate for $k=2,\ldots8$ as measured from the first qubit $Q_{1}$ to the right in the upper rows in Figs. and as a function of qubit index $k$. $J_{12}$ is $-4.9\,MHz$ and $-6.1\,MHz$ in the devices in Figs. and , respectively.](figs/J-scaling-2by8-paper-bigger-k)

\[sec:Voltage-couplings\]Couplings of the Qubits to the Voltage Drives
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Qubits are coupled to room temperature electronics for their readout and control. Readout and control signals pass through several amplification/attenuation stages as they travel through different stages in a dilution fridge. In between these stages they are carried over transmission lines like coaxial cables or the lines on a printed circuit board. We will content ourselves here with modeling this coupling mechanism simply by voltage sources driving the quantum chip through transmission lines(which we assume to be inifinite in extent 