and “drive ports”. This reduces a large portion of the design of multi-qubit superconducting devices into a classical microwave engineering problem (up to the assumptions and approximations we are making here) and allows one to avoid any numerical multi-mode block-diagonalization or fitting of electromagnetic finite-element simulations over a range of frequencies which are both expensive if not prohibitive computational procedures.

We propose the following effective Hamiltonian to desribe a multi-qubit superconducting device consisting of low anharmonicity qubits coupled to each other and to the external world through a linear passive distributed microwave structure:

$$\begin{aligned}
\hat{\mathcal{H}}/\hbar & = & \stackrel[i=1]{N}{\sum}\omega_{i}\hat{b}_{i}^{\dagger}\hat{b}_{i}+\frac{\delta_{i}}{2}\hat{b}_{i}^{\dagger}\hat{b}_{i}(\hat{b}_{i}^{\dagger}\hat{b}_{i}-1)+\underset{i,j}{\sum}J_{ij}(\hat{b}_{i}\hat{b}_{j}^{\dagger}+\hat{b}_{i}^{\dagger}\hat{b}_{j})+\stackrel[i=1]{N}{\sum}\stackrel[d=1]{N_{D}}{\sum}\varepsilon_{id}(\hat{b}_{i}-\hat{b}_{i}^{\dagger})V_{d}\nonumber \\
 & + & \underset{i,k}{\sum}\chi_{ik}\hat{b}_{i}^{\dagger}\hat{b}_{i}\hat{a}_{k}^{\dagger}\hat{a}_{k}+\stackrel[k=1]{M}{\sum}\omega_{R_{k}}\hat{a}_{k}^{\dagger}\hat{a}_{k}+\frac{\chi_{kk}^{(R)}}{2}\hat{a}_{k}^{\dagger}\hat{a}_{k}(\hat{a}_{k}^{\dagger}\hat{a}_{k}-1)+\underset{k,k'}{\sum}J_{kk'}^{\left(R\right)}(\hat{a}_{k}\hat{a}_{k'}^{\dagger}+\hat{a}_{k}^{\dagger}\hat{a}_{k'})\label{eq:Hamiltonian}\end{aligned}$$

where we have $N$ qubit modes and $M$ resonator modes represented as Duffing oscillators in the harmonic oscillator basis and $N_{D}$ voltage drives. In the first line we have terms corresponding to the qubit subspace: $\hat{b}_{i}^{(\dagger)}$ is the annihilation(creation) operator of the qubit mode $i$ of frequency $\omega_{i}$ and anharmonicity $\delta_{i}$. In the second line we have the resonator terms: $\hat{a}_{k}^{(\dagger)}$ is the annihilation(creation) operator of the resonator mode $k$ with frequency $\omega_{R_{k}}$ and anharmonicity (or self-Kerr) $\chi_{kk}^{(R)}$ (We will be using the terms “resonator” and “internal mode” interchangibly below to refer to the microwave modes 