Fig. 1. Trajectories of quantum state motion with different ${\varphi }_c$ where ${\varphi }_c\in \left[{\varphi }_i,{\varphi }_f\right]$

Fig. 2. Cost of different trajectories vary with ${\varphi }_c$

Fig. 3. Comparison of initial trajectory $p_{\mathrm{0}}$, trajectory with variational optimization $p_{\mathrm{1}}$ and trajectory with gradient descent algorithm optimization $p_{\mathrm{2}}$, and their costs are $c_{\mathrm{0}}=\mathrm{0.0598}$, $c_{\mathrm{1}}=\mathrm{0.0491}$, $c_{\mathrm{2}}=\mathrm{0.0465}$ respectively