# Optimized Unconventional Geometric Gates in Superconducting Circuits
## 3 Quantum Gates and Robustness
In the areas I, II, III, the fidelities of the quantum gates reached \(F\leq 99.95\%\), \(99.95\%<F<99.98\%\), and \(F>99.98\%\), respectively. We can see that the white curve is an increasing function as the ratio increases in \((-1,2.41)\) and is a decreasing function as the ratio decreases \((2.41,10)\). This means that the fidelity of the quantum gate increased as the ratio approached 2.41 and decreased away from 2.41. Because the Rabi frequency was a sinusoidal pulse, the ratio could not be taken as 2.41. In this article, we mainly discuss the influence of the ratio on the quantum gate. In fact, the polar angle does not necessarily have to be \(\pi/4\). For example, we took the polar angle as \(\pi/3\) and \(\pi/6\). The fidelities of the quantum gates had the same trend as a function of the ratio, and their transition point was 2 and 6.5, respectively. Here, we give the reason to explain this trend in fidelities versus ratio. As discussed above, when the quantum state evolves along the meridian, the Rabi frequency area is \(\pi/4\). The ratio of phases has an effect on the evolution of quantum states along latitude lines. Thus, we plotted the Rabi frequency area along the latitude line as a function of \(\eta\) in Figure 3. Clearly, the area of the T-gate (red line) and H-gate (blue line) was reduced when \(\eta\) approached 2.41 from the left or from the right. A smaller Rabi area with the same positive Rabi frequency \(\Omega(t)\) meant less time for the quantum system to interact with the external environment. Therefore, the decoherence noise was suppressed. Figure 3: The Rabi frequency area along the latitude line, and the change in azimuth \(\beta_{-}\) as functions of the ratio \(\eta\in(-1,10]\). Figure 2: Gate fidelities of the T-gate and H-gate as a function of the parameters \(\eta\) and \(\kappa\), (**a**) \(\eta\in(-1,2.41)\) and (**b**) \(\eta\in(2.41,10]\) for the T-gate, (**c**) \(\eta\in(-1,2.41)\) and (**d**) \(\eta\in(2.41,10]\) for the H-gate. In the areas I, II, and III, the fidelities of the quantum gates reached \(F\leq 99.95\%\), \(99.95\%<F<99.98\%\), and \(F>99.98\%\), respectively. In the following, we take the ratios of \(\eta=0,1,2,3\) as examples to test their robustness against the control errors, i.e., the qubit-frequency drift \(\delta\) and the deviation \(\epsilon\) of the driving amplitude, and compare their performance with the OSSP scheme and pure dynamical case. The detailed steps of constructing the quantum OSSP and dynamical gate are in Appendices B and C. The Hamiltonian affected by these controlled errors is written as [33; 35]  
\[H(t)=\frac{1}{2} \begin{pmatrix}-(\Delta(t)+\delta\Omega_{M})&(1+ \epsilon)\Omega(t)e^{-ip(t)}\\ (1+\epsilon)\Omega(t)e^{ip(t)}&\Delta(t)+\delta\Omega_{M}\end{pmatrix} . \tag{13}\]  
The noise \(\delta\) appears in the diagonal elements of the matrix, affecting the phase change of the qubit, and \(\epsilon\) appears in the off-diagonal elements of the matrix, affecting the flipping between qubits. According to experimental papers [36], we set the decoherence and dephasing ratio as \(\kappa_{-}=\kappa_{z}=\kappa=4\times 2\pi\) kHz. Moreover, we calculated the Rabi area of different ratios. When the ratios were zero, one, two, and three, the areas of the T-gate were \(1.34+\frac{\pi}{2},0.39+\frac{\pi}{2},0.077+\frac{\pi}{2}\), and \(0.0813+\frac{\pi}{2}\), respectively, and the areas of the H-gate were \(5.36+\frac{\pi}{2},\,\pi,0.3067+\frac{\pi}{2}\), and \(0.3253+\frac{\pi}{2}\), respectively. For the dynamical gates, they were \(5\pi/4\) for the T-gate and \(3\pi/2\) for the H-gate. Moreover, for the OSSP scheme, the areas were all \(2\pi\) for the T- and H-gate. It was found that by adjusting the ratio reasonably, the Rabi area could be made smaller than for the OSSP and the dynamic scheme.