## 2 Two Loops
### Directly Coupled Flux Qubits  
We then consider that two flux qubits, threaded by an external magnetic flux \(\Phi_{\text{ext,i}}\) with \(i=1(2)\) for left(right) qubit, are coupled through a common branch with a Josephson junction as shown in Figure 1. The periodic boundary conditions can also be obtained from the fluxoid quantization condition of superconducting loop, to yield the effective potential describing the dynamics of the system, as follows: [5]  
\[k_{1}l+\tilde{k}\bar{l}+\varphi_{1}+\bar{\varphi} =2\pi(n_{1}+f_{1}+f_{1,\text{ind}}) \tag{8}\] \[k_{2}l-\tilde{k}\bar{l}+\varphi_{2}-\bar{\varphi} =2\pi(n_{2}+f_{2}+f_{2,\text{ind}}), \tag{9}\]  
where \(k_{i}\) and \(\tilde{k}\) are the wave vector of the Cooper pairs of the left(or right) qubit branch with length \(l\) and the central branch with length \(\bar{l}\), respectively, and \(f_{i}=\Phi_{\text{ext,i}}/\Phi_{0}\). Here, \(\varphi_{i}=\varphi_{ia}+\varphi_{ib}+\varphi_{ic}\)with \(\varphi_{ia}\)'s being the phase differences of the Cooper pair wave function across the Josephson junction, \(f_{1(2),\text{ind}}\) is the induced magnetic flux of the left(right) loop, and \(n_{i}\)'s are integer. For the time being, for simplicity, we neglect the mutual inductance and then the induced flux, \(f_{\text{ind,i}}=\Phi_{\text{ind,i}}/\Phi_{0}\), for the left qubit loop is given by \(f_{\text{ind,1}}=(1/\Phi_{0})(L_{g}I_{1}+\bar{L}_{g}\bar{I})\), where \(L_{g}I_{1}\) is the induced flux due to the current \(I_{1}\) flowing the left qubit branch with the geometric self-inductance \(L_{g}\) and length \(l\) and \(\bar{L}_{g}\bar{I}\) due to the current \(\bar{I}\) flowing through the central branch with the geometric self-inductance \(\bar{L}_{g}\) and length \(\bar{I}\). Furthermore, for the right qubit loop, we have \(f_{\text{ind,2}}=(1/\Phi_{0})(L_{g}I_{2}-\bar{L}_{g}\bar{I})\), where the different sign is due to the different circularity of the central branch current with respect to the directions of the piercing fluxes \(f_{1}\) and \(f_{2}\). We introduce the kinetic inductances \(L_{K}=m_{c}l/\,An_{c}q_{c}^{2}\) and \(\bar{L}_{K}=m_{c}\bar{l}/\,An_{c}q_{c}^{2}\), and then the induced fluxes become  
\[f_{\text{ind,1}} = -\frac{L_{g}}{L_{K}}\frac{l}{2\pi}k_{1}-\frac{\bar{L}_{g}}{\bar{ L}_{K}}\frac{\bar{l}}{2\pi}\tilde{k}, \tag{10}\] \[f_{\text{ind,2}} = -\frac{L_{g}}{L_{K}}\frac{l}{2\pi}k_{2}+\frac{\bar{L}_{g}}{\bar{ L}_{K}}\frac{\bar{l}}{2\pi}\tilde{k}. \tag{11}\]  
If we assume that the loops have the same cross section \(A\) and Cooper pair density \(n_{c}\), the current conservation condition \(\bar{I}=I_{1}-I_{2}\) at the nodes of the circuit can be represented as  
\[\tilde{k}=k_{1}-k_{2}. \tag{12}\]  
From Equations (8), (9), and (12), we have  
\[k_{1,2}=\frac{\pi L_{K}}{l}\Bigg{(}\frac{1}{L_{K}+L_{g}}\pm\frac{1}{L_{K}+L_{ g}+2(\bar{L}_{K}+\bar{L}_{g})}\Bigg{)}f_{1\varphi}^{0}+\frac{\pi L_{K}}{l} \Bigg{(}\frac{1}{L_{K}+L_{g}}\mp\frac{1}{L_{K}+L_{g}+2(\bar{L}_{K}+\bar{L}_{g} )}\Bigg{)}f_{2\varphi}^{0}, \tag{13}\]  
\[\tilde{k}\,=\,k_{1}-k_{2}=\frac{2\pi L_{K}}{l}\frac{1}{L_{K}+L_{g}+2(\bar{L}_{K }+\bar{L}_{g})}\left(f_{1\varphi}^{0}-f_{2\varphi}^{0}\right) \tag{14}\]  
with  
\[f_{1\varphi}^{0}=n_{1}+f_{1}-\frac{\varphi_{1}+\bar{\varphi}}{2\pi},\,\,\,f_{ 2\varphi}^{0}=n_{2}+f_{2}-\frac{\varphi_{2}-\bar{\varphi}}{2\pi}, \tag{15}\]  
where we use the relation \(l/L_{K}=\tilde{l}/\bar{L}_{K}\) from the definition of kinetic inductances. we can see that the second term of Equation (17) is the inductive coupling energy of two-qubit current states. In this study, thus, we can find the exact expression of the inductive interaction energy \(J_{1,\text{ind}}\) in terms of the self-inductance \(\tilde{L}_{K}\) and \(\tilde{L}_{g}\), of the common branch and the mutual inductance \(L_{M}\).