# Optimized Unconventional Geometric Gates in Superconducting Circuits
## 3 Quantum Gates and Robustness
If \(\eta>2.41\), the Hamiltonian reads \(H(t)=\frac{1}{2}\Delta(t)(\ket{1}\bra{1}-\ket{0}\bra{0})+\frac{1}{2}\Omega(t)[ e^{i\beta(t)}\ket{1}\bra{0}+\text{H.c}]\) and the Rabi frequency is \(\Omega(t)=\frac{\sqrt{2}}{2}[\eta-\frac{\sqrt{2}}{2}(1+\eta)]\hat{\beta}(t)\). If and only if \(\eta=2.41\), we have \(\Omega(t)=0\). Finally, the state returns back to the north pole \((0,\pi/[2(1+\eta)(2-\sqrt{2})])\) along the geodesic \(\beta(t)=\pi/[2(1+\eta)(2-\sqrt{2})]\) for \((T_{2},T_{3})\). Accordingly, the Hamiltonian is expressed as \(H(t)=\frac{1}{2}\Omega(t)[e^{i(\beta(t)-\frac{\pi}{2})}\ket{1}\bra{0}+\text{H.c}]\) with \(\Omega(t)=-\dot{\alpha}(t)>0\). The Rabi fre  
Figure 1: **(a)** Schematic diagram of quantum state evolution; the curve is composed of two longitude lines and a section of latitude line, where \(\alpha\) and \(\beta\) are the polar and azimuth angles, respectively. **(b)** A single transmon is driven by a microwave; we can adjust the Rabi frequency through the xy control line and adjust the detuning through the z control line. quency area is also \(\pi/4\) and the solid angle is calculated as \(2\gamma_{g}(T)=\frac{\pi}{4(1+\eta)}\). Finally, the total phase is \(f(T)=(1+\eta)\gamma_{g}(T)=\frac{\pi}{8}\). The Hamiltonian along every piecewise path can be described as a two-level system and is driven by the Rabi frequency and the detuning,  
\[H(t)=\frac{1}{2} \begin{pmatrix}-\Delta(t)&\Omega(t)e^{-i\mu(t)}\\ \Omega(t)e^{i\mu(t)}&\Delta(t)\end{pmatrix} . \tag{11}\]  
The process of constructing the H-gate is very similar to that of the T-gate. For simplicity, the evolution parameters and the Rabi frequency are given directly below:  
\[(1) (\frac{\pi}{4},0)\rightarrow(0,0),\] \[(2) (0,\frac{1}{1+\eta}\frac{2\pi}{\sqrt{2}-2})\rightarrow(\frac{ \pi}{4},\frac{1}{1+\eta}\frac{2\pi}{\sqrt{2}-2}),\] \[(3) (\frac{\pi}{4},\frac{1}{1+\eta}\frac{2\pi}{\sqrt{2}-2})\rightarrow( \frac{\pi}{4},0).\]  
The Rabi frequency reads \(\Omega(t)=-\dot{a}(t)\) in \([0,T_{1}]\), and \(\Omega(t)=\dot{a}(t)\) in \((T_{1},T_{2}]\). Meanwhile, \(\Omega(t)=\frac{\sqrt{2}}{2}[\frac{\sqrt{2}}{2}(1+\eta)-\eta]\dot{\beta}(t)\) when \(\eta<2.41\), and \(\Omega(t)=\frac{\sqrt{2}}{2}[\eta-\frac{\sqrt{2}}{2}(1+\eta)]\dot{\beta}(t)\) when \(\eta>2.41\) in \((T_{2},T_{3}]\). We next take \(\Omega(t)=\Omega_{M}\sin(\frac{\pi t}{\eta})\) with \(\Omega_{M}=20\times 2\pi\) MHz and \(T_{ij}=T_{j}-T_{i}\) being the total time of every piecewise path, to evaluate the gate robustness of the implemented T-gate and H-gate in our scheme under decoherence, using the Lindblad master equation [32]  
\[\dot{\rho}=-i[H(t),\rho]+\frac{1}{2}\kappa_{z}\mathcal{L}(\mathcal{A}_{z})+ \frac{1}{2}\kappa_{-}\mathcal{L}(\mathcal{A}_{-}), \tag{12}\]  
where \(\rho\) is the density matrix of the input state, \(\mathcal{L}(\mathcal{A})=2\mathcal{A}\rho\mathcal{A}^{\dagger}-\mathcal{A}^{ \dagger}\mathcal{A}\rho-\rho\mathcal{A}^{\dagger}\mathcal{A}\) is the Lindblad operator for decay operator \(\mathcal{A}_{-}=\sum_{k=1}^{+\infty}\sqrt{k}\ket{k-1}\bra{k}\) and dephasing operator \(\mathcal{A}_{z}=\sum_{k=1}^{+\infty}k\ket{k}\bra{k}\)[33]. To test the robustness of the T- and H-gate, we set the general input initial state as \(\ket{\varphi(t)}=\cos\theta\ket{0}+\sin\theta\ket{1}\). After the action of the Lie group \(U(T)\), the final ideal state \(\ket{\varphi(T)}\) was \(\cos\theta\ket{0}+\exp(i\pi/4)\ket{1}\) and \([(\cos\theta+\sin\theta)\ket{0}+(\cos\theta-\sin\theta)\ket{1}]/\sqrt{2}\), respectively. The gate fidelity was defined as \(F_{G}=\frac{1}{2\pi}\int_{0}^{2\pi}\bra{\varphi(T)}\ket{\rho}\varphi(T)\)\(d\theta\)[34], and the integration was numerically done for the 1001 input states with \(\theta\) being uniformly distributed within \([0,2\pi]\). In Figure 2, we obtained the fidelities of the T-gate and H-gate as a function of the parameters \(\eta\in(-1,2.41)\cup(2.41,10]\) and the decoherence rate \(\kappa_{-}=\kappa_{z}=\kappa\in[0,8]\times 2\pi\)kHz.