Energy shortcut of quantum protocols by optimal control

Abstract

We introduce an energetically-optimal method inspired from Shortcut-To-Adiabaticity (STA) processes, named Quantum-Optimal-Shortcut-To-Energetics (QOSTE). QOSTE produces the same transformation as STA for a given protocol used in quantum technologies or thermodynamics, but at the lowest possible energy cost. We apply optimal control theory to analytically design the QOSTE controls for a qubit and show that the minimal energy cost is determined by the length of the geodesic in the rotating frame given by the original protocol. A numerical example in the case of a two-level quantum system under the Landau-Zener protocol illustrates the method. We observe a dramatic reduction in energy with respect to standard STA methods. Finally, using gradient-based optimization algorithms and highlighting the emerging trade-off between robustness and energy cost, we design robust QOSTE outperforming STA both in robustness and energy efficiency.

Figures (3)

• Figure 1: Representation on the Bloch sphere of the excited state of (an arbitrary) $H_0(t)$ between $t=0$ and $t=t_f$, defining the so-called adiabatic trajectory (red points) and the length $L_{0,t_f}$. The adiabatic trajectory in the rotating picture with respect to $H_0(t)$ is plotted in green points, representing the length $\tilde{L}_{0,t_f}$ (which is shown to be equal to $L_{0,t_f}$, see Supplementary Material). The geodesic in such rotating picture, $\tilde{G}_{0,t_f}$, is represented in blue dashed line.
• Figure 2: Trajectories of the qubit state when driven respectively by $H_{CD}(t) = H_0(t) + V_{CD}$ (in red), by $H(t) = H_0(t) + V_\text{QOSTE}(t)$ (in green), and by $H_0(t)$ (in blue). Inset: Amplitudes of the control functions for the QOSTE, $v_k(t) = \frac{1}{2\omega_i}{\rm Tr}[V_\text{QOSTE}(t)\sigma_k]$, $k=x$ (green dots), $k=y$ (green pluses), $k=z$ (green solid line), and for the counter-diabatic drive, $v_\text{CD}(t):= \frac{1}{2\omega_i}{\rm Tr}[V_\text{CD}(t)\sigma_y]$ (red solid line). We use $\Delta(t) = \Delta_0 +\Delta_d t/t_f$ with $\Delta_0/\omega = -10$, $\Delta_d/\omega = 20$ and $\omega t_f= 1$.
• Figure 3: Plot of the average fidelity $\bar{F}$ against the energy cost ${\cal C} = \frac{\omega_i}{2}\int_0^{t_f} dt \vec{v}^2$ (normalized with ${\cal C}[V_\text{QOSTE}]$). The uncertainty range is $\epsilon = 0.15$, and $N_\eta = 7$. The green star indicates the average fidelity of QOSTE, ${\bar{F}} \simeq 0.989$, while the red star corresponds to the one of CD-STA, ${\bar{F}}\simeq 0.987$. Inset: Plot of the fidelity with respect to the target state $|\langle e_f|\psi_\eta(t_f)\rangle|^2$ as a function of the level of uncertainty represented by $\eta$, for the CD drive (in yellow dashed), for QOSTE (in black dotted) and finally for the GRAPE-designed robust QOSTE for the respective energy costs ${\cal C}/{\cal C}[V_\text{QOSTE}] =2.71$ (in green), ${\cal C}/{\cal C}[V_\text{QOSTE}] =3.87$ (in orange ), and ${\cal C}/{\cal C}[V_\text{QOSTE}] = 8.00$ (in blue).