## 3 Three Loops
### Qubit Coupling Mediated by a Connecting Loop
With the kinetic inductance \(L_{K}^{\prime}=m_{c}l^{\prime}/An_{c}q_{c}^{2}\), the induced fluxes can be written as  
\[f_{\text{ind,1}} = -\frac{L_{g}}{L_{K}}\frac{l}{2\pi}k_{1}+\frac{L_{g}}{L_{K}}\frac{ \tilde{l}}{2\pi}\tilde{k}_{1}-\frac{\tilde{L}_{M}}{L_{K}}\frac{\tilde{l}}{2\pi} \tilde{k}_{2}-\frac{L_{M}^{\prime}}{L_{K}^{\prime}}\frac{l^{\prime}}{2\pi}(k_{ 1}^{\prime}+k_{2}^{\prime})-\frac{L_{M}^{\beta}}{L_{K}}\frac{l}{2\pi}k_{2}, \tag{23}\] \[f_{\text{ind,2}} = -\frac{L_{g}}{L_{K}}\frac{l}{2\pi}k_{2}+\frac{L_{g}}{L_{K}}\frac{ l}{2\pi}\tilde{k}_{2}-\frac{L_{M}}{L_{K}}\frac{l}{2\pi}\tilde{k}_{1}-\frac{L_{M}^{ \prime}}{L_{K}^{\prime}}\frac{l^{\prime}}{2\pi}(k_{1}^{\prime}+k_{2}^{\prime })-\frac{L_{M}^{\beta}}{L_{K}}\frac{l}{2\pi}k_{1},\] (24) \[f_{\text{ind}}^{\prime} = -\frac{L_{g}^{\prime}}{L_{K}^{\prime}}\frac{l^{\prime}}{2\pi}(k_{ 1}^{\prime}+k_{2}^{\prime})-\frac{\tilde{L}_{g}}{L_{K}}\frac{\tilde{l}}{2\pi}( \tilde{k}_{1}+\tilde{k}_{2})-\frac{L_{M}^{\alpha}}{L_{K}}\frac{l}{2\pi}(k_{1} +k_{2}), \tag{25}\]  
and then the boundary conditions become  
\[\left(1+\frac{L_{g}}{L_{K}}\right)k_{1}l-\left(1+\frac{\tilde{L} _{g}}{L_{K}}\right)\tilde{k}_{1}\tilde{l}+\frac{\tilde{L}_{M}}{L_{K}}\tilde{k }_{2}\tilde{l}+\frac{L_{M}^{\prime}}{L_{K}^{\prime}}(k_{1}^{\prime}+k_{2}^{ \prime})l^{\prime}+\frac{L_{M}^{\beta}}{L_{K}}k_{2}l =2\pi f_{1\varphi} \tag{26}\] \[\left(1+\frac{L_{g}}{L_{K}}\right)k_{2}l-\left(1+\frac{\tilde{L} _{g}}{L_{K}}\right)\tilde{k}_{2}\tilde{l}+\frac{\tilde{L}_{M}}{L_{K}}\tilde{k }_{1}\tilde{l}+\frac{L_{M}^{\prime}}{L_{K}^{\prime}}(k_{1}^{\prime}+k_{2}^{ \prime})l^{\prime}+\frac{L_{M}^{\beta}}{L_{K}}k_{1}l =2\pi f_{2\varphi}\] (27) \[\left(1+\frac{\tilde{L}_{g}}{L_{K}}\right)(\tilde{k}_{1}+\tilde{ k}_{2})\tilde{l}+\left(1+\frac{L_{g}^{\prime}}{L_{K}^{\prime}}\right)(k_{1}^{ \prime}+k_{2}^{\prime})l^{\prime}+\frac{L_{M}^{\alpha}}{L_{K}}(k_{1}+k_{2})l =2\pi f_{\varphi}^{\prime} \tag{28}\]  
with  
\[f_{1\varphi}=n_{1}+f_{1}-\frac{\varphi_{1}-\varphi_{1}}{2\pi},\ \ f_{2\varphi}=n_{2}+f_{2}-\frac{\varphi_{2}-\varphi_{2}}{2\pi},\ \ f_{\varphi}^{\prime}=n^{\prime}+f^{\prime}-\frac{\varphi^{\prime}+\tilde{ \varphi}}{2\pi}. \tag{29}\]  
The current conservation conditions at the nodes of the circuit can be represented as \(k_{1}^{\prime}=k_{2}^{\prime}=k_{1}+\tilde{k}_{1}=k_{2}+\tilde{k}_{2}\), from which we have  
\[k_{2}l-\tilde{k}_{2}\tilde{l}+\varphi_{2}-\tilde{\varphi}_{2} =2\pi(n_{2}+f_{2}+f_{2,\text{ind}})\]0 \[k_{2}l-\tilde{k}_{2}\tilde{l}+\varphi_{2}-\tilde{\varphi}_{2} =2\pi(n_{2}+f_{2}+f_{2,\text{ind}})\]1  
\[k_{2}l-\tilde{k}_{2}\tilde{l}+\varphi_{2}-\tilde{\varphi}_{2} =2\pi(n_{2}+f_{2}+f_{2,\text{ind}})\]2 Two flux qubits are coupled through a central connecting qubit loop. Here, \(L_{g}\) and \(L_{g}^{\prime}\) are self-inductances and \(L_{M}^{a}\), \(L_{M}^{\beta}\), \(\tilde{L}_{M}\) and \(L_{M}^{\prime}\) are mutual inductances. between the directly coupled central loop and qubit loops, which is similar to the interaction energy, \(\sim(\bar{L}_{K}+\bar{L}_{g}-L_{M})\), in Equation (17) for directly coupled two flux qubits. The second term describes the inductive coupling energy, \(J_{2,\text{ind}}\), between the left and right flux qubits,  
\[k_{2}l-\tilde{k}_{2}\tilde{l}+\varphi_{2}-\tilde{\varphi}_{2} =2\pi(n_{2}+f_{2}+f_{2,\text{ind}})\]3  
neglecting the kinetic inductances. This term provides the interaction energy of the two qubit current state originating from the self-inductance of the central loop, \(L_{S}^{\prime}\) and \(\bar{L}_{S}\), and the mutual inductances, \(L_{M}^{\beta},L_{M}^{\alpha},L_{M}^{\prime}\), and \(\bar{L}_{M}\). The first term in the numerator of Equation (37), \(\bar{L}_{S}^{2}\), describes the two-qubit interaction via two common branches with self-inductance, \(\bar{L}_{S}\).