# Optimized Unconventional Geometric Gates in Superconducting Circuits
## 4 Implementation on Superconducting Circuits
### Two-Qubit Gate  
We next discuss the construction of the ratio-optimized two-qubit gate on two directly capacitively coupled transmon qubits, labeled as \(T_{1}\) and \(T_{2}\), as in Figure 6. The Hamiltonian of the coupled system was written as  
\[H_{0}(t)=\sum\limits_{i=1,2}\sum\limits_{j=1}^{+\infty}\left[j\omega_{i}-\frac{ j(j-1)}{2}\alpha_{i}^{\prime}\right]\left|j\right\rangle_{i}\left\langle j \right|+g(a_{1}^{\intercal}-a_{1})(a_{2}^{\intercal}-a_{2}), \tag{17}\]  
where \(\omega_{j}\) and \(\alpha_{j}\) are the transmon's frequency and anharmonicity, respectively. In fact, the above formula is the general Jaynes-Cummings model. \(g\) is the coupling strength, which is written as \(g=C_{g}\sqrt{\omega_{1}\omega_{2}}/\sqrt{(C_{1}+C_{g})(C_{2}+C_{g})}\). Once the quantum chip is prepared, the coupling strength and qubit frequency are fixed and not adjustable. Therefore, we needed to add external degrees of freedom to achieve a tunable coupling and efficient quantum control. Specifically, we added an external magnetic field \(\epsilon(t)\) to modulate the SQUID frequency [40]. The corresponding Hamiltonian was  
\[H(t)=H_{0}(t)+f(\epsilon)(\left|1\right\rangle_{2}\left\langle 1\right|+\left|2 \right\rangle_{2}\left\langle 2\right|), \tag{18}\]  
where \(f(\epsilon)\) is the frequency response of the external field. Here, \(f(\epsilon)\) was set to \(f(\epsilon)=\hat{F}(t)\), where \(F(t)=-\beta\cos[vt+\varphi(t)]\) with \(\beta\),\(\
u\), and \(\varphi\) the amplitude, frequency, and phase, respectively. The modified qubit frequency was \(\omega_{2}(t)=\omega_{2}+\hat{F}(t)\). Then, we used  
\[U_{1} =\exp\left[-i\sum\limits_{i=1,2}\sum\limits_{j=1}^{+\infty}\left[ j\omega_{i}-\frac{j(j-1)}{2}\alpha_{i}^{\prime}\right]\left|j\right\rangle_{i} \left\langle j\right|\right]\] \[U_{2} =\exp\left[-iF(t)(\left|1\right\rangle_{2}\left\langle 1\right|+ \left|2\right\rangle_{2}\left\langle 2\right|)\right] \tag{19}\]  
to rotate the Hamiltonian picture. The new Hamiltonian was obtained as  
\[H(t)=g\Big{[}\left|10\right\rangle\left\langle 01\right|e^{i\Delta_{1}t}+\sqrt{ 2}\left|20\right\rangle\left\langle 11\right|e^{i(\Delta_{1}-a_{1}^{\prime})t}+ \sqrt{2}\left|11\right\rangle\left\langle 02\right|e^{i(\Delta_{1}+a_{2}^{ \prime})t}\Big{]}e^{i\beta\cos(vt+\varphi(t))}+\text{H.c}, \tag{20}\]  
where \(\Delta_{1}=\omega_{1}-\omega_{2}\). We next handled the exponent part with the Jacobi-Anger identity,  
\[e^{i\beta\cos\theta}=\sum\limits_{n=-\infty}^{\infty}i^{n}J_{n}(\beta)e^{in\theta}, \tag{21}\]  
in which \(J_{-m}(\beta)=(-1)^{m}J_{m}(\beta)\), and \(J_{m}(\beta)\) are Bessel functions of the first kind. Meanwhile, setting \(\
u=\Delta_{1}+a_{2}^{\prime}+\Delta^{\prime}\) to obtain an off-resonance coupling with detuning \(\Delta^{\prime}\) between qubit \(\left|11\right\rangle\) and \(\left|02\right\rangle\) and neglecting the high-order oscillating terms by rotating-wave approximation yielded the effective Hamiltonian as  
\[H=-\frac{1}{2}\Delta^{\prime}(\left|11\right\rangle\left\langle 11\right|- \left|02\right\rangle\left\langle 02\right|)+\frac{1}{2}g^{\prime}\Big{[}e^{-ip(t)} \left|11\right\rangle\left\langle 02\right|+H.c\Big{]}, \tag{22}\]  
Figure 6: Schematic diagram of two directly capacitively coupled transmon qubits, labeled as \(T_{1}\) and \(T_{2}\), where \(T_{2}\) is biased by an ac magnetic flux which periodically modulates its qubit frequency. where \(g^{\prime}=2\sqrt{2}gI_{1}(\beta)\) is the effective coupling strength that can be modified by \(\beta\), and the effective phase is \(\mu(t)=\varphi(t)-\frac{\pi}{2}\). Obviously, the Hamiltonian in Equation (22) had the same form as Equation (11). Hence, the two excited states \(\left|11\right\rangle\) would accumulate a total phase similarly to the one-qubit case, i.e., \(\left|11\right\rangle\rightarrow\exp[-if(T)]\left|11\right\rangle\).