Robust control of quantum gates via sequential convex programming

Resource trade-offs can often be established by solving an appropriate robust optimization problem for a variety of scenarios involving constraints on optimization variables and uncertainties. Using an approach based on sequential convex programming, we demonstrate that quantum gate transformations can be made substantially robust against uncertainties while simultaneously using limited resources of control amplitude and bandwidth. Achieving such a high degree of robustness requires a quantitative model that specifies the range and character of the uncertainties. Using a model of a controlled one-qubit system for illustrative simulations, we identify robust control fields for a universal gate set and explore the trade-off between the worst-case gate fidelity and the field fluence. Our results demonstrate that, even for this simple model, there exists a rich variety of control design possibilities. In addition, we study the effect of noise represented by a stochastic uncertainty model.

Figure 1

Left column: Control fields $c(t,\theta )$ that are solutions of the worst-case robust optimization problem, (19). Right column: Logarithms of corresponding distances, ${\log }_{10}\mathcal{D}(\theta ,\omega )$, as functions of ${\omega }_x\in [0.99,1.01]$ and ${\omega }_z\in [1.8,2.2]$. Results are shown for eight combinations of $N$ and $T$ for each of the three target gates of Eq. (17): (a) the identity gate, (b) the Hadamard gate, and (c) the phase gate.

Figure 2

The logarithm of the worst-case distance, ${\log }_{10}{\mathcal{D}}_{\mathrm{wc}}\left(\theta \right)$, versus the fluence $\Phi \left(\theta \right)$ for control fields that are solutions of the optimization problem, (21), with $W=W_{\mathrm{I}}$, $T=2$, and $N=10$. Five data series, denoted by shape and color, correspond to five uncertainty sets ${\Delta }_m$ of Eqs. (20), as indicated in the legend.

Figure 3

The logarithm of the average distance, ${\log }_{10}{\mathcal{D}}_{\mathrm{avg}}\left(\theta \right)$, versus $\tau /T$, where $\tau$ is the noise filter timeconstant from Eq. (27). The control field used here is a solution of the optimization problem, (21), with $W=W_{\mathrm{I}}$, $T=2$, $N=10$, ${\Delta }_1$ of Eq. (20a), and no fluence constraint ($\gamma =\infty$). The distance is averaged under the noise process described by Eqs. (24, 25, 26, 27) with ${\overline{\omega }}_z=2$ and two $\sigma$ values: $\sigma =0.001$ [lower (red) line and circles] and $\sigma =0.02$ [upper (blue) line and circles]. The weak noise approximation (solid lines) computed using Eqs. (29) and (30) is in a very good agreement with simulated data (circles) obtained using random sampling from the noise distribution according to Eq. (28).