$10\,GHz$ for the circuit in Fig. with the following set of parameter values $g_{1}=g_{2}=100\,MHz$, $\omega_{1}=2\pi(4.90\,GHz)$ and $\omega_{2}=2\pi\left(5.10\,GHz\right)$, $\delta_{1}=\delta_{2}=-340\,MHz$. Vertical axis is $J$-coupling rate in $MHz$.](figs/fig1-J)

Example 2: Scaling of $J$ coupling rates in a multi-qubit device
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![\[fig:2x8-device\]2x8 device connectivity: 16 qubits are arranged in 2 rows. Nodes respresent the qubits while edges linking the qubits represent buses. There are two qubits connected to each bus and there is a total of $22$ buses.](figs/2by8_2QbyBUS)

![\[fig:2x8-device-4QbyBus\]2x8 device with four qubits per bus arrangement. Scaling of the $J$ couplings over the lattice is compared to the arrangement in Fig. . Crossed links represent bus resonators each connected to 4 qubits and there are $7$ buses in total.](figs/2by8_4QbyBUS)

In this section we apply the impedance formula in Eq. for the exchange couplings $J_{ij}$ to the multi-qubit device shown in Fig. to calculate the decay of $J$ over the chip. This is a simplified model of an actual multi-qubit device recently released by IBM in its online cloud environment for quantum computing: IBM Q Experience [@IBM-Q-Experience]. The device consists of $16$ qubits arranged in two rows and connected to each other by $22$ bus resonators with two qubits per bus. To compare we also 