Universally Robust Quantum Control

We study the robustness of the evolution of a quantum system against small uncontrolled variations in parameters in the Hamiltonian. We show that the fidelity susceptibility, which quantifies the perturbative error to leading order, can be expressed in superoperator form and use this to derive control pulses that are robust to any class of systematic unknown errors. The proposed optimal control protocol is equivalent to searching for a sequence of unitaries that mimics the first-order moments of the Haar distribution, i.e., it constitutes a 1-design. We highlight the power of our results for error-resistant single- and two-qubit gates.

Figure 1

Universally robust control for single-qubit gates. (a) Optimized control functionals as a function of the total evolution time $t_f$ for target-only control (gray, circles), target and robustness to a known $V$ (blue, squares), and target and robustness to an unknown $V$ (orange, triangles). (b),(c) Gate fidelity as a function of perturbation strength $\lambda$ for the cases where $V={\sigma }_z$ and $V=\overrightarrow{n}·\overrightarrow{\sigma }$ with $\overrightarrow{n}$ a random unit vector (results shown correspond to the average fidelity over 20 realizations). Lower panels shows an enlargement of the data of the infidelity $1-F$ in log-log scale. (d) Optimal control fields $\varphi (t)$ obtained for each case. We choose a target $U_{\text{target}}=\exp (-i{\sigma }_z\pi /2)$, $N_P=40$ control parameters, a balanced functional $w=1$, and an operation time $\mathrm{\Omega }{t}_f/(2\pi )=3.5$ for (b), (c), and (d).

Figure 2

Universally robust control for two-qubit gates. Plots show the gate fidelity of the perturbed evolution $H_0(t)+\lambda V$, where $H_0(t)$ is the control Hamiltonian of Eq. (16). Different curves correspond to different types of optimization procedures: target only (nonrobust, gray circles), target and robustness to a fixed $V=S_x$ (blue squares), target and robustness to all single-body operators (green crosses), target and universal robustness (orange triangles). The lower row shows the infidelity $1-F_U$ for each case. Each column shows the performance of each of the four optimized solutions under the evolution $H=H_0(t)+\lambda V$; (a) $V=S_x$, (b) $V$ is random 1-body operator, (c) $V$ is a random arbitrary operator. Evolution time in all cases is $\beta t_f/(2\pi )=5$, and $N_P=50$ control parameters are used. Results shown are averages over 20 instances.