Robust quantum control using hybrid pulse engineering

The development of efficient algorithms that generate robust quantum controls is crucial for the realization of quantum technologies. The commonly used gradient-based optimization algorithms are limited by their sensitivity to the initial guess, which affects their performance. Here we propose combining the gradient method with the simulated annealing technique to formulate a hybrid algorithm. Our numerical analysis confirms its superior convergence rate. Using the hybrid algorithm, we generate spin-selective $\pi$ pulses and employ them for experimental measurement of local noise spectra in a three-qubit nuclear magnetic resonance system. Moreover, here we describe a general method to construct noise-resilient quantum controls by incorporating noisy fields within the optimization routine of the hybrid algorithm. Upon experimental comparison with similar sequences obtained from standard algorithms, we find remarkable robustness of the hybrid sequences against dephasing errors.

Figure 1

Comparison of quantum control algorithms with respect to convergence efficiency and robustness against noise.

Figure 2

Threshold-based acceptance function (${\mathrm{\Delta }}^i$) plotted versus temperature $T^i$ and the fidelity variation $\delta {\mathrm{\Phi }}^i$. Note that ${\mathrm{\Delta }}^i$ is bounded by $[-1,0]$.

Figure 3

Illustrating the progress of the SAGRAPE algorithm in bumpy parameter space. The green and white backgrounds respectively depict the regions where GRAPE and SA contribute toward minimization. GRAPE relies on monotonic convergence, while SA allows jumping over local maxima. As the algorithm proceeds, the temperature parameter of SA is gradually reduced to keep the solution trapped in the global minimum by restraining any further jumps.

Figure 4

Flowchart for the SAGRAPE algorithm. Here $i\le \kappa$ and $j\le K$ indicate iterations over SA and SAGRAPE, respectively.

Figure 5

Average infidelity $\langle 1-{\mathrm{\Phi }}^i\rangle$ in the $i\mathrm{th}$ iteration of SAGRAPE versus computational time taken by GRAPE and SAGRAPE-$\kappa$ algorithms for (a) state control (preparing pseudo-singlet-triplet order) and (b) gate control (preparing cnot gate) respectively. Each SAGRAPE-$\kappa$ data point corresponds to $\kappa$ number of SA iterations followed by one GRAPE iteration. Here each curve is obtained by averaging over five different trials starting from the random initial guess $\left\{{\omega }_{\alpha }^0\left(j\right)\right\}$. In each trial, the initial guess was kept the same across all the different cases.

Figure 6

(a) Molecular structure of BTFBz showing three ${}^{19}\mathrm{F}$ nuclei and the table of resonance offsets ${\mathrm{\Omega }}_k/\left(2\pi \right)$ (diagonal elements) and indirect spin-spin coupling constants $J_{kl}$ (off-diagonal elements) in Hz. (b) Experimentally obtained local noise spectrum for each ${}^{19}\mathrm{F}$ spin.

Figure 7

(a) Molecular structure of TCP, its reference spectrum (labeled thermal), and its LLS spectrum. (b) The NMR pulse sequence for preparing, storing, and detecting the singlet order. A pulsed-field-gradient (PFG) with amplitude randomly varying with time is used to induce phase noise.

Figure 8

(a) The RF components ${\omega }_x\left(j\right)$ (blue solid lines, in rad/s) and ${\omega }_y\left(j\right)$ (red dashed lines, in rad/s) corresponding to the three control sequences obtained with GRAPE, GRAPE with CPMG, and RSAGRAPE-5 with CPMG plotted versus time. The latter two sequences are integrated with six CPMG $\pi$ pulses, each of amplitude $+9941$ rad/s, but their corresponding plots are cropped at $\pm 1000$ rad/s for the visibility purpose. (b) The experimentally observed singlet order versus the maximum gradient strength $G_z$ (in $\%$) for the three control sequences shown in panel (a).