Comparing and correcting robustness metrics for quantum optimal control

TL;DR

This work analyzes robustness of quantum control under quasi-static coherent errors by comparing three first-order robustness formulations: toggling $\mathcal{E}$, universal $\mathcal{E}_U$, and adjoint $\mathcal{E}_V$, revealing significant numerical differences from discretization. It introduces a critical discretization correction to the toggling estimator to align it with the true susceptibility $\mathcal{E}$ and demonstrates that direct, constrained trajectory optimization better handles fidelity and hardware constraints than indirect methods, with physics-informed robustness outperforming universal robustness in Hadamard and iSWAP tasks. The framework is validated on single- and two-qubit gates and is supported by open-source software Piccolo.jl, enabling practical adoption across quantum hardware platforms. This work thus provides both methodological guidance for robust quantum control and tools to implement it in real devices.

Abstract

Control pulses that nominally optimize fidelity are sensitive to routine hardware drift and modeling errors. Robust quantum optimal control seeks error-insensitive control pulses that maintain fidelity thresholds and obey hardware constraints. Distinct numerical approximations to the first-order error susceptibility include adjoint end-point and toggling-frame approaches. Although theoretically equivalent, we provide a novel, systematic study demonstrating important numerical differences between these two approaches. We also introduce a critical discretization correction to the widely-used toggling-frame robustness estimator, measurably improving its estimate of first-order error susceptibility. We accomplish our study by positioning robustness as a first-class objective within direct, constrained optimal control. Our approach uniquely handles control and fidelity constraints while cleanly isolating robustness for dedicated optimization. In both single- and two-qubit examples under realistic constraints, our approach provides an analytic edge for obtaining precise, physics-informed robustness.

Figures (7)

• Figure 1: (a) Qubit trajectory for a Hadamard gate without robust control. The lighter shades show deviations as the coherent error, $\epsilon Z$, increases. (b) Robustness metrics [Sec. \ref{['sec:robustness']}]: $\mathcal{E}$ is the first-order error susceptibility, $\mathcal{E}_V$ is the adjoint objective, $\mathcal{E}_T^{(j)}$ is the (corrected) toggling objective. (c) i. Robust control of a Hadamard gate under the same coherent error as (a). ii. The bar chart compares the different robustness metrics. Ideal $\mathcal{E}$ (dashed line) is met by the adjoint objective (orange), but the toggling objective (green) requires a fourth order correction, $\mathcal{E}_T^{(4)}$.
• Figure 2: The cumulative distribution function (CDF) of the maximum control (a) velocity [(b) acceleration] over solver iterations for three trajectory optimization algorithms. The red vertical line marks the constraint. Constrained indirect (C.I.) spends more time near the boundary than constrained direct (C.D.), while the unconstrained indirect (U.I) optimization occasionally exceeds the boundary. At the final solver iteration, the maximum control (c) velocity [(d) acceleration] is below the constraint boundary.
• Figure 3: (a) Solver convergence rates (measured using the adjoint objective, $\mathcal{E}_V$) are comparable for constrained indirect and constrained direct optimizers under control constraints. (b) Adding a final fidelity constraint does not impact the convergence rate of the constrained direct optimizer, but the constrained indirect optimizer is unable to converge.
• Figure 4: (a) The toggling objective [(b) adjoint objective] is optimized using a weight $Q$. The dashed black line is the first-order susceptibility of the optimized solution. Increasing $Q$ prioritizes robustness over other regularization terms, so susceptibility should decrease. In (a), the desired first-order susceptibility saturates at large $Q$ because $\mathcal{E} \ne \mathcal{E}_T^{(0)}$.
• Figure 5: The optimizer minimizes the objective $\mathcal{E}_T^{(0)}$ for a $32$ ns Hadamard gate with an increasing number of knot points ($N$=32, 64, 128, 256). The black dashed line is the first-order susceptibility of the solution, $\mathcal{E}$. The adjoint objective (orange line) is computed from the solution and matches $\mathcal{E}$, while the toggling objective (blue diamonds) only matches as the order $j$ in the $\Delta t$ expansion is increased. Higher orders are needed when $\Delta t$ is large ($N$ is small).