# Optimized Unconventional Geometric Gates in Superconducting Circuits
## 4 Implementation on Superconducting Circuits
### One-Qubit Gate  
We started by constructing one-qubit gates in superconducting circuits. Because the Josephson junction is a nonlinear inductor, the energy spectrum of the transmon qubit can be equivalent to a two-level system, i.e., \(H=\omega_{1}\sigma_{z}/2\), where \(\omega_{1}\) is the transmon's resonance frequency. The two lowest levels were used as our logical qubit. Then, we added a microwave field to drive a single transmon, as in Figure 1b. The Hamiltonian read  
\[H(t)=\sum_{k=1}^{\infty}\Bigl{[}k\omega_{1}-\frac{1}{2}k(k-1)\alpha^{\prime} \ket{k}\bra{k}\Bigr{]}+\bigl{[}\frac{1}{2}\Omega(t)a\dot{e}^{i[\int_{0}^{t} \omega(t^{\prime})\,dt^{\prime}-\mu(t)]}+H.c\bigr{]} \tag{14}\]  
where \(\omega_{1}\), \(\alpha^{\prime}\), \(\omega\), and \(\mu\) are the qubit frequency, transmon's anharmonicity, microwave frequency, and phase, respectively. Here, the annihilation operator was always associated with the emission of photons, because we ignored the process of absorbing photons using the rotation wave approximation. The second excited state \(\ket{2}\) was our main source of leakage, because of the anharmonicity of the transmon. Therefore, we truncated the annihilation operator to \(a=\ket{0}\bra{1}+\sqrt{2}\ket{1}\bra{2}\). After rotating the Hamiltonian to the new framework with \(U=U_{2}U_{1}\), where \(U_{1}=\exp{[-i\omega_{1}at]}\), and \(U_{2}=\exp{[i\int_{0}^{t}\Delta(t^{\prime})\,dt^{\prime}(\ket{1}\bra{1}-\ket{ 0}\bra{0}+3\ket{2}\bra{2})/2]}\), the new Hamiltonian was obtained as  
\[H(t) =\frac{1}{2}\{\Delta(t)(\ket{1}\bra{1}-\ket{0}\bra{0}+3\ket{2} \bra{2})\}-\alpha^{\prime}\ket{2}\bra{2} \tag{15}\] \[+\frac{1}{2}\{\Omega(t)e^{-i\mu(t)}(\ket{0}\bra{1}+\sqrt{2}\ket{1 }\bra{2})+\text{H.c}\}\]  
where \(\Delta(t)=\omega_{1}-\omega(t)\). As in Figure 1b, the Rabi frequency could be adjusted by the xy control line and arbitrary waveform generator. The Z control line modified the detuning by changing the magnetic flux in the superconducting quantum interference device (SQUID). Figure 4: Test of the robustness of the \(\eta=0,1,2,3\) T-gate and H-gate against control errors and comparison with the OSSP scheme and dynamical gates. The results show that we can improve the fidelity of the gate by choosing a specific ratio. (**a**–**c**) \(\delta\) and (**b**–**d**) \(\epsilon\) errors. We used the derivative removal via adiabatic gate (DRAG) technology to suppress the qubit leakage error caused by state \(|2\rangle\)[38]. The corrected rabbi frequency was  
\[\Omega_{D}(t)=\Omega(t)-\{i\Omega(t)+[\hat{\mu}(t)+\Delta(t)]\Omega(t)\}/2 \alpha^{\prime} \tag{16}\]  
The detailed derivation of the DRAG correction waveform is shown in Appendix D. According to a current state-of-art experiment [39], we set the decoherence rate and anharmonicity of the transmon as \(\kappa=\kappa_{-}=\kappa_{z}=4\times 2\pi\)kHz and \(\alpha^{\prime}=2\pi\times 220\) MHz, respectively. Generally speaking, a strong Rabi frequency \(\Omega(t)\) corresponds to a fast rotation speed. However, the coupling term \(|1\rangle\)\(\langle 2|\) in the Hamiltonian did not allow us to arbitrarily increase \(\Omega(t)\). Therefore, it was necessary for us to further discuss the choice of Rabbi frequency \(\Omega(t)\). In the top subfigure of Figure 5, we plotted the fidelities of the T-gate ans H-gate as a function of the amplitude \(\Omega_{M}\). As expected, the relationship between quantum gates fidelities and pulse strength was fluctuating. We calculated the peak of the curve, which corresponded to the highest fidelity and optimized Rabbi frequency. When \(\Omega_{M}\) was taken as \(\Omega_{M}=31\times 2\pi\) MHz, the fidelity of the T-gate reached the highest \(F_{T}^{\rm{G}}\cong 99.98\%\), and the highest fidelity of the H-gate was \(F_{H}^{\rm{G}}\cong 99.95\%\) when \(\Omega_{M}=12\times 2\pi\) MHz. Let us consider a concrete example.