Shortcuts to adiabaticity with a quantum control field

Abstract

Quantum adiabatic dynamics is the crucial element of adiabatic quantum computing and quantum annealing. Shortcuts to adiabaticity enable acceleration of the computational time by suppressing unwanted non-adiabatic processes with designed classical fields. Here, we consider quantum state transfer in the Landau-Zener model, which exemplifies the key elements of quantum adiabatic dynamics. We argue that non-adiabatic transitions can be suppressed by autonomous quantum dynamics, which involves coupling the Landau-Zener qubit to a second quantum system. By tuning the coupling strength, the composite quantum dynamics can reduce the probability of unwanted processes by more than two orders of magnitude. This is a prime example of control where the quantum properties of the control fields are key for implementing shortcuts to adiabaticity.

Figures (17)

• Figure 1: Interference-assisted shortcut to adiabaticity: The ultra-strong coupling of a LZ qubit to a quantum field (bosonic or fermionic) leads to high-fidelity adiabatic passage, even for fast sweeps where the isolated LZ dynamics would be in the diabatic regime. Here, $x_0$ scales the strength of the coupling, and $\omega_c$ is the eigenfrequency of the quantum field.
• Figure 2: Instantaneous energy branches $E_\pm^{(\pm)}(t)$ of the qubit-field Hamiltonian $\hat{H}$ of Eq. \ref{['eq:H']} (solid lines) for exemplary values of $x_0$ and $\omega_c$ and protocol time window $t_f=10g/\epsilon$. Times outside this window, $t>t_f$, are shaded gray. Dashed lines are the qubit-field system instantaneous eigenenergies at $x_0=0$. Subfigures exemplify Regime (I), where the coupling with the field tends to close the minimal gap (subplot (a), $x_0=\omega_c=0.6g$); Regime (II), where the coupling with the field tends to promote high-fidelity transfer (Subplots (b) and (c)). In (b) the coupling increases the gap at $t=0$ and flattens the spectrum in its vicinity ($x_0=\omega_c=1.8g$). In (c) "side gaps" (indicated by gray dotted lines) occur at $t_\mathrm{sg}\approx \pm 3.8g/\epsilon$ ($x_0=1.5g$, $\omega_c=4g$). Subplot (d) illustrates Regime (III), where the anticrossings at $t_\mathrm{sg}\approx 12g/\epsilon>t_f$ occur outside the protocol time window ($x_0=4g$, $\omega_c=12g$). The symbols on top help to locate the corresponding spectrum in the phase diagram of Fig. \ref{['fig:4']}.
• Figure 3: Evolution of the transition probability $P(t)$, Eq. \ref{['eq:transition_prob']}, and purity $\gamma(t)$, Eq. \ref{['eq:purity']}, for the qubit-field dynamics governed by Hamiltonian \ref{['eq:H']} and in the diabatic regime, $\epsilon = 2g^2$. Three different choices of $x_0$ and $\omega_c$ are shown, illustrating the regimes of the annealing dynamics: (I) $x_0 = 0.1g$ and $\omega_c = x_0$, (II) $x_0 = 1.2g$ and $\omega_c = 0.6x_0$, (III) $x_0 = 4g$ and $\omega_c = 3x_0$. The black dashed line represents the bare LZ dynamics ($x_0=0$). The qubit-field system is initialized at time $t_i = -10g/\epsilon$ in state $\vert\Psi\rangle_0 = \vert -\rangle_{t_i}\otimes\vert\downarrow\rangle$ and evolved to time $t_f = -t_i$. Extending the time window damps the oscillations and further reduces $P(t)$ in regimes (II) and (III); see SM SMref.
• Figure 4: Protocol's efficiency: (a) Infidelity $\mathcal{I}=P(t_f)$\ref{['eq:transition_prob']} and (b) impurity $1-\overline{\gamma}=1-\gamma(t_f)$\ref{['eq:purity']} of the final state as a function of the parameters $x_0/g$ and $\omega_c/x_0$ for $t_f\approx 10g/\epsilon$. Annealing procedure and choice of parameters are as in Fig. \ref{['fig:3']}. The dashed lines indicate the critical values $\Delta_c^{(1)}= 2g$ and $\Delta_c^{(2)}$, separating the three regimes. The dotted curve splits regime (II) into two subregions: Below the curve there is only a single anticrossing at $t=0$. Above the curve new anticrossings emerge at $t=\pm|t_\mathrm{sg}|$. Markers in (a) refer to the corresponding spectra in Fig. \ref{['fig:2']}. For the chosen annealing time the infidelity oscillates at the end of the protocol. The values we report are obtained by taking the local minimum of $P(t_f)$ and $\gamma(t_f)$ in the window $t_f\in[8g/\epsilon,10g/\epsilon]$. The results are consistent with other optimization procedures, such as taking the mean of $P(t)$ and $\gamma(t)$ over the final 10% of the annealing window. They are also insensitive to the spectator's initial state. See SM SMref for further details.
• Figure S1: (a)-(d) (Left): Dynamics of the transition probability $P(t)$ of the Landau-Zener system after tracing out the quantum field. Solid red curves denote results for finite system-spectator couplings, $x_0\neq0$. The dashed blue curve illustrates the case where the system and spectator qubit are decoupled, $x_0=0$, and is only shown for reference. Parameter values are specified above the respective subfigures. We consistently fix $g=1$. (a)-(d) (Right): Energy branches $E_n(t)$, $n=1,\,2,\,3,\,4$, of the full system. The branches are colored according to the instantaneous population of each eigenstate, where the population of the eigenstate $\vert n(t)\rangle$ is computed as $P_n(t)=\vert \langle n(t)\vert\Psi\rangle_t\vert^2$. Dashed gray bands denote the decoupled case $(x_0=0)$. Dotted black vertical lines indicate the times at which the side gaps occur, as predicted by Eq. \ref{['eq:gap_times']}. (a) $\omega_c<g$, since $g=1$. Side gaps vanish, and we observe an effective renormalization of the central gap. (b) $\omega_c>g$, $t_\mathrm{sg}<t_f$, and $x_0/g\gg1$. The conditions on the spectator frequency and final anneal time for the side gaps to appear are satisfied; however, the strong coupling results in oscillations leading to transitions out of the system's instantaneous ground state and the population of the highest energy branch. (c) The coupling $x_0$ is now of the order of $g$. Additional anticrossings lead to energy branches corresponding to the instantaneous ground state $\vert-\rangle_t$ being populated, reducing the infidelity. (d) The coupling $x_0$ is small relative to $g$. Diabatic transitions occur at the second side gap, resulting in population transfer out of the branches coinciding with the instantaneous ground state $\vert-\rangle_t$ of the system.