Galvanic Phase Coupling of Superconducting Flux Qubits

Abstract

We investigate the galvanic coupling schemes of superconducting flux qubits. From the fundamental boundary conditions, we obtain the effective potential of the coupled system of two or three flux qubits to provide the exact Lagrangian of the system. While usually the two-qubit gate has been investigated approximately, in this study we derive the exact inductive coupling strength between two flux qubits coupled directly and coupled through a connecting central loop. We observe that the inductive coupling strength needs to be included exactly to satisfy the criteria of fault-tolerant quantum computing.

1. Introduction

Superconducting qubits have been intensively studied with the advantages of scalability and flexibility of circuit design. Noisy intermediate-scale quantum computing [1] is expected to be realized in the near future due to the remarkable advancement in qubit coherence and control. The scalable design of quantum computing requires the implementation of the two-qubit gate. The superconducting flux qubit has several possibilities of coupling two qubits, which makes the flux qubit a promising candidate for quantum computing.

In this study, we analyze the effect of the inductive coupling energy to provide the exact expression of the inductive coupling strength in terms of the geometric self-inductance and mutual inductance of the qubit loops. We, first, consider the case that two flux qubits are directly coupled through a common central branch [2]. We, then, introduce the system where three flux qubits are coupled [3,4]. The three-qubit coupled system can provide a tunable two-qubit coupling by using the central qubit loop as a coupling element. The central loop can provide a tunable two-qubit coupling parameterized by the phase differences between the Josephson junctions in the left and right qubits [5,6,7,8,9,10,11,12,13,14]. In addition to the flux qubit coupling, for charge based qubits the three loop structure gives rise to the tunable coupling for the transmons or Xmons. An external magnetic flux in the central loop controls the coupling for Xmon [15,16], and the Josephson ring modulator takes the role of the central loop for coupling two transmons [17]. This three-qubit scheme can also be extended to an array of coupled flux qubits [18] to simulate a one-dimensional chain model.

From the fundamental boundary conditions for two flux qubits coupled directly and coupled through a connecting central loop, we obtain the effective potential of the system to derive an exact Lagrangian of the coupled system with an analytic expression of the inductive coupling strength between two qubits in terms of the geometric self-inductance of the central branch or loop and the mutual inductance between two qubit loops. By calculating the Controlled NOT(CNOT) gate fidelity numerically we observe that the exact inductive coupling energy is required to satisfy the criteria of the fault-tolerant quantum computing.

2. Two Loops

2.1. Capacitively-Shunted Model for an rf-SQUID

We first consider the capacitively-shunted model for an radio frequency superconducting quantum interference device(rf-SQUID) loop, which has a Josephson junction with the critical current $I_c$ and a capacitor with capacitance C in parallel. The current flowing through the rf-SQUID loop is given by [19]

$$\begin{array}{c}\hfill I=-I_{c}sin\varphi +C\dot{V},\end{array}$$

(1)

where $\varphi$ is the superconducting phase difference across the Josephson junction. Here, the Cooper pair current I is represented by $I=-(n_{c}A{q}_c/m_c)\hslash k$ with the Cooper pair density $n_c$, the cross section A of the loop, $q_c=2e$, and $m_c=2m_e$. Using the Josephson voltage-phase relation $V=-({\mathrm{\Phi }}_0/2\pi )\dot{\varphi }$ with the superconducting unit flux quantum ${\mathrm{\Phi }}_0=h/2e$, the current relation of Equation (1) becomes

$$\begin{array}{c}\hfill -\frac{{\mathrm{\Phi }}_0^2}{2\pi L_K}\frac{l}{2\pi }k=-E_{J}sin\varphi -C{\left(\frac{{\mathrm{\Phi }}_0}{2\pi }\right)}^2\ddot{\varphi },\end{array}$$

(2)

where we introduce the kinetic inductance, $L_K=m_{c}l/A{n}_c{q}_c^2$ [5,20,21], and the Josephson coupling energy, $E_J=I_c{\mathrm{\Phi }}_0/2\pi$.

From the Lagrange equation $(d/dt)\partial \mathcal{L}/\partial \dot{\varphi }-\partial \mathcal{L}/\partial \varphi =0$ with the Lagrangian

$$\begin{array}{c}\hfill \mathcal{L}=\frac{C}{2}{\left(\frac{{\mathrm{\Phi }}_0}{2\pi }\right)}^2{\dot{\varphi }}^2-U_{\mathrm{eff}}\left(\left\{\varphi \right\}\right)\end{array}$$

(3)

and the effective potential of the system $U_{\mathrm{eff}}\left(\varphi \right)$, we can obtain the equation of motion of the phase variable of a Josephson junction, $C{({\mathrm{\Phi }}_0/2\pi )}^2\ddot{\varphi }=-\partial U_{\mathrm{eff}}/\partial \varphi$, and thus the relation,

$$\begin{array}{c}\hfill \frac{{\mathrm{\Phi }}_0^2}{2\pi L_K}\frac{l}{2\pi }k-E_{J}sin\varphi =-\frac{\partial U_{\mathrm{eff}}}{\partial \varphi }.\end{array}$$

(4)

The fluxoid quantization condition for a thin superconducting loop with the total threading magnetic flux ${\mathrm{\Phi }}_t$ is given by $-{\mathrm{\Phi }}_t+(m_c/q_c)\oint {\overrightarrow{v}}_c·d\overrightarrow{l}=n{\mathrm{\Phi }}_0$ [19,22], with the average velocity of Cooper pairs ${\overrightarrow{v}}_c$, which can be represented as a periodic boundary condition, $k_n\mathit{l}=2\pi n+2\pi f_t$, where $k_n$ is the wave vector of the Cooper pairs, l the circumference of the loop, and $f_t\equiv {\mathrm{\Phi }}_t/{\mathrm{\Phi }}_0$. Here, $k_{n}l$ is the phase accumulated along the circumference of the loop. For an rf-SQUID loop, the boundary condition with the phase difference $\varphi$ across the Josephson junction becomes $k_n\mathit{l}=2\pi n+2\pi f_t-\varphi$. The total flux ${\mathrm{\Phi }}_t={\mathrm{\Phi }}_{\mathrm{ext}}+{\mathrm{\Phi }}_{\mathrm{ind}}$ with the external flux ${\mathrm{\Phi }}_{\mathrm{ext}}$ and induced flux ${\mathrm{\Phi }}_{\mathrm{ind}}=L_{g}I$ can be written by $f_t=f+f_{\mathrm{ind}}$ with $f\equiv {\mathrm{\Phi }}_{\mathrm{ext}}/{\mathrm{\Phi }}_0,f_{\mathrm{ind}}\equiv {\mathrm{\Phi }}_{\mathrm{ind}}/{\mathrm{\Phi }}_0=L_{g}I/{\mathrm{\Phi }}_0$ and the geometric self-inductance $L_g$ of the rf-SQUID loop, resulting in the relation

$$\begin{array}{c}\hfill \left(1+\frac{L_g}{L_K}\right)k_{n}l=2\pi n+2\pi f-\varphi .\end{array}$$

(5)

From the relations of Equations (4) and (5), we obtain

$$\begin{array}{c}\hfill -\frac{\partial U_{\mathrm{eff}}}{\partial \varphi }=\frac{{\mathrm{\Phi }}_0^2}{2\pi (L_K+L_g)}\left(n+f-\frac{\varphi }{2\pi }\right)-E_{J}sin\varphi .\end{array}$$

(6)

Hence, the relation in Equation (2) describes the dynamics of a particle with kinetic energy $E_C=Q^2/2C$ with $Q=C({\mathrm{\Phi }}_0/2\pi )\dot{\varphi }$ in an effective potential, [23]

$$\begin{array}{c}\hfill U_{\mathrm{eff}}\left(\varphi \right)=\frac{{\mathrm{\Phi }}_0^2}{2(L_K+L_g)}{\left(n+f-\frac{\varphi }{2\pi }\right)}^2-E_{J}cos\varphi .\end{array}$$

(7)

2.2. Directly Coupled Flux Qubits

We then consider that two flux qubits, threaded by an external magnetic flux ${\mathrm{\Phi }}_{\mathrm{ext},i}$ with $i=1\left(2\right)$ 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]

$$\begin{array}{ccc}\hfill & & k_{1}l+\tilde{k}\tilde{l}+{\phi }_1+\tilde{\phi }=2\pi (n_1+f_1+f_{1,\mathrm{ind}})\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill & & k_{2}l-\tilde{k}\tilde{l}+{\phi }_2-\tilde{\phi }=2\pi (n_2+f_2+f_{2,\mathrm{ind}}),\hfill \end{array}$$

(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 $\tilde{l}$, respectively, and $f_i={\mathrm{\Phi }}_{\mathrm{ext},\mathrm{i}}/{\mathrm{\Phi }}_0$. Here, ${\phi }_i={\phi }_{ia}+{\phi }_{ib}+{\phi }_{ic}$ with ${\phi }_{i\alpha }$’s being the phase differences of the Cooper pair wave function across the Josephson junction, $f_{1\left(2\right),\mathrm{ind}}$ 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, $f_{\mathrm{ind},\mathrm{i}}={\mathrm{\Phi }}_{\mathrm{ind},\mathrm{i}}/{\mathrm{\Phi }}_0$, for the left qubit loop is given by $f_{\mathrm{ind},1}=(1/{\mathrm{\Phi }}_0)(L_g{I}_1+{\tilde{L}}_g\tilde{I})$, 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 ${\tilde{L}}_g\tilde{I}$ due to the current $\tilde{I}$ flowing through the central branch with the geometric self-inductance ${\tilde{L}}_g$ and length $\tilde{l}$. Furthermore, for the right qubit loop, we have $f_{\mathrm{ind},2}=(1/{\mathrm{\Phi }}_0)(L_g{I}_2-{\tilde{L}}_g\tilde{I})$, 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 $L_K=m_{c}l/A{n}_c{q}_c^2$ and ${\tilde{L}}_K=m_c\tilde{l}/A{n}_c{q}_c^2$, and then the induced fluxes become

$$\begin{array}{ccc}\hfill f_{\mathrm{ind},1}& =& -\frac{L_g}{L_K}\frac{l}{2\pi }{k}_1-\frac{{\tilde{L}}_g}{{\tilde{L}}_K}\frac{\tilde{l}}{2\pi }\tilde{k},\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill f_{\mathrm{ind},2}& =& -\frac{L_g}{L_K}\frac{l}{2\pi }{k}_2+\frac{{\tilde{L}}_g}{{\tilde{L}}_K}\frac{\tilde{l}}{2\pi }\tilde{k}.\hfill \end{array}$$

(11)

If we assume that the loops have the same cross section A and Cooper pair density $n_c$, the current conservation condition $\tilde{I}=I_1-I_2$ at the nodes of the circuit can be represented as

$$\begin{array}{c}\hfill \tilde{k}=k_1-k_2.\end{array}$$

(12)

From Equations (8), (9), and (12), we have

$$\begin{array}{ccc}\hfill k_{1,2}& =& \frac{\pi L_K}{l}\left(\frac{1}{L_K+L_g}\pm \frac{1}{L_K+L_g+2({\tilde{L}}_K+{\tilde{L}}_g)}\right)f_{1\phi }^0+\frac{\pi L_K}{l}\left(\frac{1}{L_K+L_g}\mp \frac{1}{L_K+L_g+2({\tilde{L}}_K+{\tilde{L}}_g)}\right)f_{2\phi }^0,\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill \tilde{k}& =& k_1-k_2=\frac{2\pi L_K}{l}\frac{1}{L_K+L_g+2({\tilde{L}}_K+{\tilde{L}}_g)}(f_{1\phi }^0-f_{2\phi }^0)\hfill \end{array}$$

(14)

with

$$\begin{array}{c}\hfill f_{1\phi }^0=n_1+f_1-\frac{{\phi }_1+\tilde{\phi }}{2\pi },f_{2\phi }^0=n_2+f_2-\frac{{\phi }_2-\tilde{\phi }}{2\pi },\end{array}$$

(15)

where we use the relation $l/L_K=\tilde{l}/{\tilde{L}}_K$ from the definition of kinetic inductances.

Similarly to the rf-SQUID case, we can find the effective potential satisfying the relation,

$$\begin{array}{c}\hfill \frac{{\mathrm{\Phi }}_0^2}{2\pi L_K}\frac{l}{2\pi }{k}_i-E_{J}sin{\varphi }_i=-\frac{\partial U_{\mathrm{eff}}}{\partial {\varphi }_i},\end{array}$$

(16)

with $k_i$ and ${\phi }_i$ in Equation (13) and also with $\tilde{k}$ and $\tilde{\phi }$ in Equation (14). If we consider the mutual inductance $L_M$ between two qubit loops, we can also obtain $k_i$’s in a similar manner [see Appendix A]. For this general case, thus, we can construct the effective potential which satisfies this relation in Equation (16) as

$$\begin{array}{ccc}\hfill U_{\mathrm{eff}}\left(\left\{{\phi }_i\right\}\right)& =& \frac{{\mathrm{\Phi }}_0^2}{4}\left(\frac{1}{L_K+L_g+L_M}+\frac{1}{L_K+L_g-L_M+2({\tilde{L}}_K+{\tilde{L}}_g)}\right)\left[{\left(n_1+f_1-\frac{{\phi }_1+\tilde{\phi }}{2\pi }\right)}^2+{\left(n_2+f_2-\frac{{\phi }_2-\tilde{\phi }}{2\pi }\right)}^2\right]\hfill \\ & +& {\mathrm{\Phi }}_0^2\frac{{\tilde{L}}_K+{\tilde{L}}_g-L_M}{(L_K+L_g+L_M)[L_K+L_g-L_M+2({\tilde{L}}_K+{\tilde{L}}_g)]}\left(n_1+f_1-\frac{{\phi }_1+\tilde{\phi }}{2\pi }\right)\left(n_2+f_2-\frac{{\phi }_2-\tilde{\phi }}{2\pi }\right)\hfill \\ & -& E_J\sum _{i=1}^2\sum _{\alpha =a,b,c}cos{\phi }_{i\alpha }-{\tilde{E}}_{J}cos\tilde{\phi }.\hfill \end{array}$$

(17)

One can easily check that the relation in Equation (16) can be satisfied with this effective potential and the wave vectors in Equations (A5) and (A6) in the Appendix. In usual experimental situations, the geometric inductance is dominant over the kinetic inductance $L_g\gg L_K$, [24]; thus, we can neglect the kinetic inductances.

If we consider the limit that the central branch shrinks to a point, $\tilde{l}\to 0$ and thus ${\tilde{L}}_K,{\tilde{L}}_g\to 0$, while two loops share the Josephson junction with phase difference $\tilde{\phi }$, and, further, we neglect the mutual inductance, the effective potential becomes

$$\begin{array}{ccc}\hfill U_{\mathrm{eff}}& =& \frac{{\mathrm{\Phi }}_0^2}{2L_g}\left({f_{1\phi }^0}^2+{f_{2\phi }^0}^2\right)-E_J\sum _{i=1}^2\sum _{\alpha =a,b,c}cos{\phi }_{i\alpha }-{\tilde{E}}_{J}cos\tilde{\phi },\hfill \end{array}$$

(18)

which is the usual effective potential for two flux qubits without inductive interaction. The first term is the inductive energy of a two qubit loop and can be re-expressed as $(1/2)L_g(I_1^2+I_2^2)$ with the current $I_i=({\mathrm{\Phi }}_0/L_g)f_{i\phi }^0$ of the qubit loop. For the general case, 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 $J_{1,\mathrm{ind}}$ in terms of the self-inductance ${\tilde{L}}_K$ and ${\tilde{L}}_g$, of the common branch and the mutual inductance $L_M$. Neglecting the small kinetic inductances, the two-qubit inductive interaction energy is written as

$$\begin{array}{c}\hfill J_{1,\mathrm{ind}}\approx {\mathrm{\Phi }}_0^2\frac{{\tilde{L}}_g-L_M}{(L_g+L_M)(L_g-L_M+2{\tilde{L}}_g)}{f}_{1\phi }^0{f}_{2\phi }^0.\end{array}$$

(19)

Here, the typical values are ${\tilde{L}}_g/L_g\sim$ 1/3 and $L_M/L_g\sim 0.1$ [25], and we numerically obtain $J_{1,\mathrm{ind}}/h\approx 0.2$ GHz with $E_J/h=100$ GHz [26]. If one of the loops shrinks so that the loop area becomes zero, we have ${\tilde{L}}_g\approx L_M$, and thus the contributions of the inductances ${\tilde{L}}_g$ and $L_M$ become canceled.

3. Three Loops

3.1. Qubit Coupling Mediated by a Connecting Loop

In a quantum computing scheme the two-qubit interaction should be switched on/off; moreover, the coupling strength should be controllable. Hence, the schemes introducing a central connecting loop to couple the phases of the Josephson junctions in the left and right qubit have been investigated not only for the phase-based qubit such as flux qubits [5,6,7,8,9,10,11,12,13,14] but also for the charge-based qubit such as Xmons [15,16] and transmons [17]. Figure 2 shows the prototype of this phase coupling scheme, where the central qubit has the role of the coupling loop, and the two-qubit coupling can be controlled by adjusting the piercing flux $f^{\prime }$. This scheme has been studied in an approximate way, such that the inductive energy of the loop current is neglected for very small qubit loops. In this study, we obtain the exact effective potential for general case, providing an analytic form of the inductive coupling energy between two flux qubits.

The periodic boundary conditions obtained from the fluxoid quantization condition of superconducting loop can be written as [5]

$$\begin{array}{ccc}\hfill & & k_{1}l-{\tilde{k}}_1\tilde{l}+{\phi }_1-{\tilde{\phi }}_1=2\pi (n_1+f_1+f_{1,\mathrm{ind}})\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill & & k_{2}l-{\tilde{k}}_2\tilde{l}+{\phi }_2-{\tilde{\phi }}_2=2\pi (n_2+f_2+f_{2,\mathrm{ind}})\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill & & k_1^{\prime }{l}^{\prime }+{\tilde{k}}_1\tilde{l}+k_2^{\prime }{l}^{\prime }+\tilde{\phi }+{\phi }^{\prime }=2\pi (n^{\prime }+f^{\prime }+f_{\mathrm{ind}}^{\prime }),\hfill \end{array}$$

(22)

where ${\phi }_i={\phi }_{ia}+{\phi }_{ib}+{\phi }_{ic}$ with $i=1,2$, ${\phi }^{\prime }={\phi }_1^{\prime }+{\phi }_2^{\prime }$ and $\tilde{\phi }={\tilde{\phi }}_1+{\tilde{\phi }}_2$. The induced magnetic flux, $f_{\mathrm{ind},\mathrm{i}}={\mathrm{\Phi }}_{\mathrm{ind},\mathrm{i}}/{\mathrm{\Phi }}_0$, for the left qubit loop is given by $f_{\mathrm{ind},1}=(1/{\mathrm{\Phi }}_0)[L_g{I}_1-{\tilde{L}}_g{\tilde{I}}_1+{\tilde{L}}_M{\tilde{I}}_2+L_M^{\prime }(I_1^{\prime }+I_2^{\prime })+L_M^{\beta }{I}_2]$, where the first two terms are the same as the terms for the directly coupled qubits. In Figure 2, ${\tilde{L}}_g$ and $L_g^{\prime }$ are the self-inductance of the branches with length $\tilde{l}$ and $l^{\prime }$ of the central loop, respectively. $L_M^{\alpha }$ is the geometric mutual inductance between the central loop and the right qubit branch with length l and $L_M^{\beta }$ between the left qubit loop and the right qubit branch. ${\tilde{L}}_M$ and $L_M^{\prime }$ are the mutual inductances between the left qubit loop and the branches of central loop with lengths $\tilde{l}$ and $l^{\prime }$, respectively. For the right qubit loop the same mutual inductances can be introduced. As shown in Figure 2, the third term ${\tilde{L}}_M{\tilde{I}}_2$ is the induced flux due to the current ${\tilde{I}}_2$ flowing in a central branch and the mutual inductance ${\tilde{L}}_M$ between the left qubit loop and the central branch with length $\tilde{l}$, the fourth term $L_M^{\prime }(I_1^{\prime }+I_2^{\prime })$ due to the current $I_1^{\prime }$ and $I_2^{\prime }$ flowing central branches and the mutual inductance $L_M^{\prime }$ between the left qubit loop and a central branch with length $l^{\prime }$, and the last term $L_M^{\beta }{I}_2$ due to the current $I_2$ in the right qubit branch and the mutual inductance $L^{\beta }$ between the left qubit loop and the right qubit branch with length l. The other induced fluxes are also given in a similar manner as $f_{\mathrm{ind},2}=(1/{\mathrm{\Phi }}_0)[L_g{I}_2-{\tilde{L}}_g{\tilde{I}}_2+{\tilde{L}}_M{\tilde{I}}_1+L_M^{\prime }(I_1^{\prime }+I_2^{\prime })+L_M^{\beta }{I}_1]$, and $f_{\mathrm{ind}}^{\prime }=(1/{\mathrm{\Phi }}_0)[L_g^{\prime }(I_1^{\prime }+I_2^{\prime })+{\tilde{L}}_g({\tilde{I}}_1+{\tilde{I}}_2)+L_M^{\alpha }(I_1+I_2)]$.

With the kinetic inductance $L_K^{\prime }=m_c{l}^{\prime }/A{n}_c{q}_c^2$, the induced fluxes can be written as

$$\begin{array}{ccc}\hfill f_{\mathrm{ind},1}& =& -\frac{L_g}{L_K}\frac{l}{2\pi }{k}_1+\frac{{\tilde{L}}_g}{{\tilde{L}}_K}\frac{\tilde{l}}{2\pi }{\tilde{k}}_1-\frac{{\tilde{L}}_M}{{\tilde{L}}_K}\frac{\tilde{l}}{2\pi }{\tilde{k}}_2-\frac{L_M^{\prime }}{L_K^{\prime }}\frac{l^{\prime }}{2\pi }(k_1^{\prime }+k_2^{\prime })-\frac{L_M^{\beta }}{L_K}\frac{l}{2\pi }{k}_2,\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill f_{\mathrm{ind},2}& =& -\frac{L_g}{L_K}\frac{l}{2\pi }{k}_2+\frac{{\tilde{L}}_g}{{\tilde{L}}_K}\frac{\tilde{l}}{2\pi }{\tilde{k}}_2-\frac{{\tilde{L}}_M}{{\tilde{L}}_K}\frac{\tilde{l}}{2\pi }{\tilde{k}}_1-\frac{L_M^{\prime }}{L_K^{\prime }}\frac{l^{\prime }}{2\pi }(k_1^{\prime }+k_2^{\prime })-\frac{L_M^{\beta }}{L_K}\frac{l}{2\pi }{k}_1,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill f_{\mathrm{ind}}^{\prime }& =& -\frac{L_g^{\prime }}{L_K^{\prime }}\frac{l^{\prime }}{2\pi }(k_1^{\prime }+k_2^{\prime })-\frac{{\tilde{L}}_g}{{\tilde{L}}_K}\frac{\tilde{l}}{2\pi }({\tilde{k}}_1+{\tilde{k}}_2)-\frac{L_M^{\alpha }}{L_K}\frac{l}{2\pi }(k_1+k_2),\hfill \end{array}$$

(25)

and then the boundary conditions become

$$\begin{array}{c}\hfill \left(1+\frac{L_g}{L_K}\right)k_{1}l-\left(1+\frac{{\tilde{L}}_g}{{\tilde{L}}_K}\right){\tilde{k}}_1\tilde{l}+\frac{{\tilde{L}}_M}{{\tilde{L}}_K}{\tilde{k}}_2\tilde{l}+\frac{L_M^{\prime }}{L_K^{\prime }}(k_1^{\prime }+k_2^{\prime })l^{\prime }+\frac{L_M^{\beta }}{L_K}{k}_{2}l=2\pi f_{1\phi }\end{array}$$

(26)

$$\begin{array}{c}\hfill \left(1+\frac{L_g}{L_K}\right)k_{2}l-\left(1+\frac{{\tilde{L}}_g}{{\tilde{L}}_K}\right){\tilde{k}}_2\tilde{l}+\frac{{\tilde{L}}_M}{{\tilde{L}}_K}{\tilde{k}}_1\tilde{l}+\frac{L_M^{\prime }}{L_K^{\prime }}(k_1^{\prime }+k_2^{\prime })l^{\prime }+\frac{L_M^{\beta }}{L_K}{k}_{1}l=2\pi f_{2\phi }\end{array}$$

(27)

$$\begin{array}{c}\hfill \left(1+\frac{{\tilde{L}}_g}{{\tilde{L}}_K}\right)({\tilde{k}}_1+{\tilde{k}}_2)\tilde{l}+\left(1+\frac{L_g^{\prime }}{L_K^{\prime }}\right)(k_1^{\prime }+k_2^{\prime })l^{\prime }+\frac{L_M^{\alpha }}{L_K}(k_1+k_2)l=2\pi f_{\phi }^{\prime }\end{array}$$

(28)

with

$$\begin{array}{c}\hfill f_{1\phi }=n_1+f_1-\frac{{\phi }_1-{\tilde{\phi }}_1}{2\pi },f_{2\phi }=n_2+f_2-\frac{{\phi }_2-{\tilde{\phi }}_2}{2\pi },f_{\phi }^{\prime }=n^{\prime }+f^{\prime }-\frac{{\phi }^{\prime }+\tilde{\phi }}{2\pi }.\end{array}$$

(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

$$\begin{array}{ccc}\hfill k_1^{\prime }+k_2^{\prime }& =& k_1+k_2+{\tilde{k}}_1+{\tilde{k}}_2\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill k_1-k_2& =& -({\tilde{k}}_1-{\tilde{k}}_2)\hfill \end{array}$$

(31)

We can obtain one equation by summing Equations (26) and (27) and the other from Equation (28) in conjunction with Equation (30), resulting in

$$\left(\begin{array}{cc}\left(1+\frac{{\tilde{L}}_g}{{\tilde{L}}_K}-\frac{{\tilde{L}}_M}{{\tilde{L}}_K}\right)\tilde{l}-\frac{2L_M^{\prime }}{L_K^{\prime }}{l}^{\prime }& -\left(1+\frac{L_g}{L_K}+\frac{L_M^{\beta }}{L_K}\right)l-\frac{2L_M^{\prime }}{L_K^{\prime }}{l}^{\prime }\\ \left(1+\frac{L_g^{\prime }}{L_K^{\prime }}\right)l^{\prime }+\left(1+\frac{{\tilde{L}}_g}{{\tilde{L}}_K}\right)\tilde{l}& \left(1+\frac{L_g^{\prime }}{L_K^{\prime }}\right)l^{\prime }+\frac{L_M^{\alpha }}{L_K}l\end{array}\right)\left(\begin{array}{c}{\tilde{k}}_1+{\tilde{k}}_2\\ k_1+k_2\end{array}\right)=2\pi \left(\begin{array}{c}-f_{1\phi }-f_{2\phi }\\ f_{\phi }^{\prime }\end{array}\right)$$

(32)

from which we can calculate $k_1+k_2$ and ${\tilde{k}}_1+{\tilde{k}}_2$. Further, from the equation obtained by subtracting Equation (27) from (26) and the equation of Equation (31), we can calculate $k_1-k_2$ and $-({\tilde{k}}_1-{\tilde{k}}_2)$. As a result, we can obtain $k_i$’s as

$$\begin{array}{ccc}\hfill k_{1,2}& =& \frac{2\pi }{l}\frac{L_K}{2}\left(\frac{L_K^{\prime }+L_g^{\prime }+{\tilde{L}}_K+{\tilde{L}}_g}{L_1^2}\pm \frac{1}{{\tilde{L}}_K+{\tilde{L}}_g+{\tilde{L}}_M+L_K+L_g-L_M^{\beta }}\right)f_{1\phi }\hfill \\ & +& \frac{2\pi }{l}\frac{L_K}{2}\left(\frac{L_K^{\prime }+L_g^{\prime }+{\tilde{L}}_K+{\tilde{L}}_g}{L_1^2}\mp \frac{1}{{\tilde{L}}_K+{\tilde{L}}_g+{\tilde{L}}_M+L_K+L_g-L_M^{\beta }}\right)f_{2\phi }\hfill \\ & +& \frac{2\pi }{l}\frac{L_K}{2}\frac{{\tilde{L}}_K+{\tilde{L}}_g-{\tilde{L}}_M-2L_M^{\prime }}{L_1^2}{f}_{\phi }^{\prime }\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill {\tilde{k}}_{1,2}& =& \frac{2\pi }{l}\frac{L_K}{2}\left(-\frac{L_K^{\prime }+L_g^{\prime }+L_M^{\alpha }}{L_1^2}\mp \frac{1}{{\tilde{L}}_K+{\tilde{L}}_g+{\tilde{L}}_M+L_K+L_g-L_M^{\beta }}\right)f_{1\phi }\hfill \\ & +& \frac{2\pi }{l}\frac{L_K}{2}\left(-\frac{L_K^{\prime }+L_g^{\prime }+L_M^{\alpha }}{L_1^2}\pm \frac{1}{{\tilde{L}}_K+{\tilde{L}}_g+{\tilde{L}}_M+L_K+L_g-L_M^{\beta }}\right)f_{2\phi }\hfill \\ & +& \frac{2\pi }{l}\frac{L_K}{2}\frac{{\tilde{L}}_K+{\tilde{L}}_g+L_M^{\beta }+2L_M^{\prime }}{L_1^2}{f}_{\phi }^{\prime }\hfill \end{array}$$

(34)

$$\begin{array}{ccc}\hfill k_1^{\prime }& =& k_2^{\prime }=\frac{2\pi }{l}\frac{L_K}{2}\frac{{\tilde{L}}_K+{\tilde{L}}_g-L_M^{\alpha }}{L_1^2}(f_{1\phi }+f_{2\phi })+\frac{2\pi }{l}\frac{L_K}{2}\frac{{\tilde{L}}_K+{\tilde{L}}_g-{\tilde{L}}_M+L_K+L_g+L_M^{\beta }}{L_1^2}{f}_{\phi }^{\prime },\hfill \end{array}$$

(35)

where $L_1^2\equiv ({\tilde{L}}_K+{\tilde{L}}_g+{\tilde{L}}_M-2L_M^{\prime })(L_K^{\prime }+L_g^{\prime }+L_M^{\alpha })+(L_K+L_g+L_M^{\beta }+2L_M^{\prime })(L_K^{\prime }+L_g^{\prime }+{\tilde{L}}_K+{\tilde{L}}_g)$.

Now, we consider the case that the three loops in Figure 2 are identical, and thus the mutual inductance, $L_M^{\alpha }$, between the central loop and the right branch with length l is the same as that between the left qubit loop and the three branches of the central loop, i.e., $L_M^{\alpha }={\tilde{L}}_M+2L_M^{\prime }$. We then can construct the effective potential $U_{\mathrm{eff}}\left(\left\{{\phi }_i\right\}\right)$ for this case of identical loops, satisfying the relation in Equation (16) with the wave vectors in Equations (33)–(35), as

$$\begin{array}{ccc}\hfill U_{\mathrm{eff}}\left(\left\{{\phi }_i\right\}\right)& =& \frac{{\mathrm{\Phi }}_0^2}{4}\left(\frac{L_K^{\prime }+L_g^{\prime }+{\tilde{L}}_K+{\tilde{L}}_g}{L_1^2}+\frac{1}{{\tilde{L}}_K+{\tilde{L}}_g+{\tilde{L}}_M+L_K+L_g-L_M^{\beta }}\right)\left({f_{1\phi }}^2+{f_{2\phi }}^2\right)\hfill \\ & +& \frac{{\mathrm{\Phi }}_0^2}{2}\left(\frac{L_K^{\prime }+L_g^{\prime }+{\tilde{L}}_K+{\tilde{L}}_g}{L_1^2}-\frac{1}{{\tilde{L}}_K+{\tilde{L}}_g+{\tilde{L}}_M+L_K+L_g-L_M^{\beta }}\right)f_{1\phi }{f}_{2\phi }\hfill \\ & +& \frac{{\mathrm{\Phi }}_0^2}{4}\frac{{\tilde{L}}_K+{\tilde{L}}_g-{\tilde{L}}_M+L_K+L_g+L_M^{\beta }}{L_1^2}{f_{\phi }^{\prime }}^2+\frac{{\mathrm{\Phi }}_0^2}{2}\frac{{\tilde{L}}_K+{\tilde{L}}_g-L_M^{\alpha }}{L_1^2}{f}_{\phi }^{\prime }\left(f_{1\phi }+f_{2\phi }\right)\hfill \\ & -& E_J\sum _{i=1}^2\sum _{\alpha =a,b,c}cos{\phi }_{i\alpha }-{\tilde{E}}_J\sum _{i=1}^{2}cos{\tilde{\phi }}_i-E_J^{\prime }\sum _{i=1}^{2}cos{\phi }_i^{\prime }.\hfill \end{array}$$

(36)

Here, the first and third terms show the inductive energies of the qubit loops and the central loop, respectively. The fourth term describes the interaction energy, $\sim ({\tilde{L}}_K+{\tilde{L}}_g-L_M^{\alpha })$, between the directly coupled central loop and qubit loops, which is similar to the interaction energy, $\sim ({\tilde{L}}_K+{\tilde{L}}_g-L_M)$, in Equation (17) for directly coupled two flux qubits.

The second term describes the inductive coupling energy, $J_{2,\mathrm{ind}}$, between the left and right flux qubits,

$$\begin{array}{c}\hfill J_{2,\mathrm{ind}}\approx \frac{{\mathrm{\Phi }}_0^2}{2}\frac{{\tilde{L}}_g^2+L_M^{\alpha }(2L_M^{\prime }-{\tilde{L}}_M)-2L_M^{\beta }(L_g^{\prime }+{\tilde{L}}_g)-{\tilde{L}}_g(2L_M^{\prime }-{\tilde{L}}_M+L_M^{\alpha })}{L_1^2(L_g+{\tilde{L}}_g+{\tilde{L}}_M-L_M^{\beta })}{f}_{1\phi }{f}_{2\phi },\end{array}$$

(37)

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, $L_g^{\prime }$ and ${\tilde{L}}_g$, and the mutual inductances, $L_M^{\beta },L_M^{\alpha },L_M^{\prime }$, and ${\tilde{L}}_M$. The first term in the numerator of Equation (37), ${\tilde{L}}_g^2$, describes the two-qubit interaction via two common branches with self-inductance, ${\tilde{L}}_g$. The second term in the numerator, $L_M^{\alpha }(2L_M^{\prime }-{\tilde{L}}_M)$, represents the two-qubit indirect interaction via the mutual inductances, $L_M^{\alpha }$, between the left (right) branch and the central loop and then the mutual inductance $2L_M^{\prime }-{\tilde{L}}_M$ between the central loop and the right (left) qubit loop. The third term in the numerator, $2L_M^{\beta }(L_g^{\prime }+{\tilde{L}}_g)$, describes the direct interaction between two qubit loops through the mutual inductance, $L_M^{\beta }$. The last term in the numerator shows the combined two-qubit interaction consisting of the self-inductance, ${\tilde{L}}_g$, of the central loop and the mutual inductance, $2L_M^{\prime }-{\tilde{L}}_M+L_M^{\alpha }$, between a qubit loop and a central branch. In the parameter regime similar to that for the direct coupling [25,26], we numerically obtain the inductive coupling energy $J_{2,\mathrm{ind}}/h\approx$ 0.06 GHz, which is smaller than $J_{1,\mathrm{ind}}/h\approx$ 0.2 GHz for direct coupling, because, in this case, the coupling is mediated indirectly via an intervening loop.

3.2. CNOT Gate Fidelity

The controlled-NOT (CNOT) gate with the single qubit gate provides a universal set for quantum computing. For the CNOT gate operation, we consider a kind of cross-resonance scheme that two qubits in Figure 1 and Figure 2 are driven by an oscillating magnetic field resonant with the target qubit frequency. When two qubits (left and right qubits) are coupled, the Hamiltonian for coupled qubits can be written as [27]

$$\begin{array}{c}\hfill H=E_z^l{\sigma }_z\otimes I+I\otimes E_z^r{\sigma }_z-t_q^l{\sigma }_x\otimes I-t_q^{r}I\otimes {\sigma }_x+J{\sigma }_z\otimes {\sigma }_z,\end{array}$$

(38)

where $E_z^l=h^l/2+gcos\omega t$, and $E_z^r=h^r/2+gcos\omega t$ with the static qubit energy gap $h^{l\left(r\right)}$ for left and right qubits and the coupling constant g between the qubit and the oscillating magnetic field with frequency $\omega =2t_q^r$.

The fidelity for CNOT gate operation is given by $F\left(t\right)=\mathrm{Tr}\left(M\left(t\right)M_{\mathrm{CNOT}}\right)/4$, [28] where $M_{\mathrm{CNOT}}$ is the matrix for the perfect CNOT operation, and $M\left(t\right)$ is the truth table amplitude at time t. At the degeneracy point where $h^l=h^r$, we introduce qubit states in a transformed coordinate as $|0\rangle =\left(\right|\downarrow \rangle +|\uparrow \rangle )/\sqrt{2}$ and $|1\rangle =\left(\right|\downarrow \rangle -|\uparrow \rangle )/\sqrt{2}$. $M_{\mathrm{CNOT}}$ is represented as

$$\begin{array}{c}\hfill M_{\mathrm{CNOT}}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right],\end{array}$$

(39)

where, if we consider the first qubit as the control qubit and the second the target qubit, $M_{\mathrm{CNOT}}$ shows that for the control qubit state $|1\rangle$ the target qubit state becomes flipped, while the target qubit state remains unchanged for the control qubit state $|0\rangle$, i.e., $|00\rangle \to |00\rangle ,|01\rangle \to |01\rangle ,|10\rangle \to |11\rangle$ and $|11\rangle \to |10\rangle$. $M{\left(t\right)}_{ij}=P_{|i\rangle \to |j\rangle }$ with $|i\rangle ,|j\rangle \in \left\{\right|00\rangle ,|01\rangle ,$$|10\rangle ,|11\rangle \}$ is the probability that the state $|i\rangle$ evolves to the state $|j\rangle$ at time t, which can be calculated by using the Hamiltonian in Equation (38) [27]. In

Table 1. Fidelities for CNOT gate operation where $F_0$ is calculated with the given value of J in the Table, while $F_1$ and $F_2$ are calculated with $J_1/h=J/h-0.2$ GHz and $J_2/h=J/h-0.06$ GHz, respectively.

J/h (GHz)	0.5	1.0	2.0	3.0
$F_0$	0.9921	0.9977	0.9978	0.9980
$F_1$	0.6597	0.9613	0.9974	0.9979
$F_2$	0.9390	0.9975	0.9978	0.9980

, we show the fidelity for the CNOT gate operation, where $F_0$ is the fidelity with the given coupling energy J consisting of both the Josephson junction energy and the inductive coupling energy. If we neglect the inductive coupling energy, $J_{1,\mathrm{ind}}/h=0.2$ GHz in Figure 1 and $J_{2,\mathrm{ind}}/h=0.06$ GHz in Figure 2, the fidelities $F_1$ and $F_2$ can be calculated with the coupling energies $J_1=J-J_{1,\mathrm{ind}}$ and $J_2=J-J_{2,\mathrm{ind}}$, respectively. For a weak two-qubit coupling regime, $J/h\sim$ 0.5 GHz, the fidelities $F_1$ and $F_2$ deviate severely from $F_0$ as shown in Table 1. Even for the strong coupling regime, $J/h\gtrsim 1$ GHz, we can observe in Table 1 that the contribution of inductive coupling should be exactly included to satisfy the criteria of the fidelity error, $\delta F\sim {10}^{-4}$, of the fault tolerant quantum computing.

4. Conclusions

We studied the galvanic coupling schemes for two superconducting flux qubits. From the fundamental boundary conditions we obtained the exact Lagrangian of the system to derive the inductive coupling strength between two qubits coupled directly through a common branch and coupled through a central intervening loop. While the two-qubit gate with flux qubits has been investigated in an approximate way, the present study considered the inductive energies exactly through the geometric mutual inductance as well as the self-inductance to provide an analytic form of the inductive coupling energy. We numerically calculated the CNOT gate fidelity to show that for even strong two-qubit coupling as well as weak coupling the inductive coupling energy must be taken into account accurately.