Superadiabatic control of quantum operations

Adiabatic pulses are used extensively to enable robust control of quantum operations. We introduce an approach to adiabatic control that uses the superadiabatic quality factor as a performance metric to design robust, high-fidelity pulses. This approach permits the systematic design of quantum control schemes to maximize the adiabaticity of a unitary operation in a particular time interval given the available control resources. The interplay between adiabaticity, fidelity, and robustness of the resulting pulses is examined for the case of single-qubit inversion, and superadiabatic pulses are demonstrated to have improved robustness to control errors. A numerical search strategy is developed to find a broader class of adiabatic operations, including multiqubit adiabatic unitaries. We illustrate the utility of this search strategy by designing control waveforms that adiabatically implement a two-qubit entangling gate for a model NMR system.

Figure 1

A tanh/tan pulse of length $\tau =120 \mu \mathrm{s}$ was optimized by varying $A, \kappa$, and $\xi$ in Eqs. (6) and (7) and setting ${\omega }_1^{\max }$ to 80 krad/s. The resulting $Q_s$-optimized pulse is compared to the $Q_1$-optimal tanh/tan pulse. In (a), the first ten adiabatic $Q$ factors, defined in Eq. (3), are plotted for both pulses on a natural log scale. We compare the performance of these two pulses by systematically reducing the pulse length $\tau$. The infidelity of the inversion for each pulse length $\tau$ is plotted in (b). The quantum state's trajectory for the $Q_1$-optimized pulse is plotted in (c), and the trajectory of the $Q_s$-optimized pulse is plotted in (d). The angles ${\alpha }_1\left(t\right)$ and ${\alpha }_s\left(t\right)$ are plotted as a function of time for the $Q_1$ ($s=2$) and the $Q_s$ ($s=5$) pulses in (e) and (f), respectively.

Figure 2

(a) The infidelity of two $Q_s$-optimized pulses are compared with the $Q_1$-optimized pulse as a function of pulse length. The length of a hard $\pi$ pulse at ${\omega }_1=80$ krad/s is indicated by the vertical dotted line. The dashed black line plots the infidelity of the hard pulse as the pulse length is reduced to zero. The inset shows a magnified version of the plot in the range from 80 to 150 $\mu \mathrm{s}$ to show the improved performance of the pulse optimized for 120 $\mu \mathrm{s}$ at longer times. (b) $Q_s$ as a function of pulse length for each of the optimized pulses plotted in (a). The inset shows the behavior of $Q$ as the pulse length approaches zero.

Figure 3

The evolutionary strategy is applied to a guess pulse to optimize $Q_1$ and $Q_2$ sequentially. The infidelity of the resulting pulse, plotted as a solid line, shows improvement over the $Q_s$-optimal tanh/tan pulse of Fig. 2.

Figure 4

Robustness of four optimized pulses to $B_0$ and $B_1$ inhomogeneity. Pulse performance is examined at two pulse lengths, 46 and 120 $\mu \mathrm{s}$. The hard $\pi$-pulse fidelity is plotted as a dotted line. In (a), $\delta \omega \left(t\right)$ was subjected to a constant additive offset ranging from $-140$ to 140 krad/s. In (b), ${\omega }_1\left(t\right)$ was subjected to a multiplicative offset ranging from 0 to 3. The pulses are more robust at the longer pulse length of 120 $\mu \mathrm{s}$. The superadiabatic pulses offer improved robustness with respect to both types of offsets.

Figure 5

Fidelity profiles of the two-qubit entangling pulse that takes the state $|{\psi }_i\rangle =|00\rangle$ to the state $|{\psi }_t\rangle =\frac{1}{\sqrt{2}}(|00\rangle +|11\rangle )$. The evolutionary algorithm was applied to a linear guess pulse (dotted line) for one round of optimization (dash-dotted line) and three rounds (solid line), where a round of optimization consists of each point in the pulse serving as the center of perturbation. A comparison is made to a diabatic gate that creates the same target state (dashed line). The minimum time of the diabatic gate given $J=209$ Hz is plotted as a vertical dotted line. Inset: $Q_1\left(t\right)$ is plotted for the three pulses, showing the algorithm's improvement in the first adiabatic $Q$ factor.

Figure 6

Eigenvalues of the optimized two-qubit entangling Hamiltonian versus time. The adiabatic theorem requires that energy levels not cross, which is satisfied here. The eigenvalues corresponding to the trajectory between $|00\rangle$ at $t=0$ and the Bell state at $t=\tau$ are plotted as a solid line. The eigenvalues are plotted in units of $\hslash /s$, using $2\alpha =64$ krad/s, $2\beta =57$ krad/s, and $A=78.5$ krad/s.

Figure 7

The evolutionary strategy used to search for optimally adiabatic rf pulses, consisting of a four-step iterative procedure. (a) The center of perturbation is chosen. (b) The radius is chosen, defining an interval of perturbation. (c) The curve is perturbed, and (d) the best perturbation is kept.

Figure 8

$Q_1$ of the guess pulse for different choices of $\mathrm{\Delta }{\omega }^C\left(0\right)$ (carbon, system $A$) and $\mathrm{\Delta }{\omega }^H\left(0\right)$ (hydrogen, system $B$). Lighter colors correspond to more adiabatic pulses. The black arrow indicates the guess pulse that was chosen. The color bar indicates the value of $Q_1$ for a pulse length of 10 ms.

Figure 9

The linear guess pulse (black dotted line) and two $Q_1$-optimized pulses, obtained by applying the evolutionary algorithm with each point as the center of perturbation once (blue dash-dotted line) or three times (red solid line).