Probing the Weak-Driving Quantum Speed Limit via Drift-Aware Shooting Methods

Abstract: A central goal of quantum optimal control is to achieve high-fidelity and low-energy control pulses. When quantum optimal control methods optimize every point of a pulse discretized over small time steps independently this can yield high fidelity control but also results in broadband and energy-hungry waveforms. We extend MAGICARP -- a shooting method inspired by Pontryagin's maximum principle on energy that generates an entire pulse from a small set of parameters, making it smooth and energy-efficient by construction -- from driftless systems to closed systems with the constant drift Hamiltonian of two exchange-coupled spins in an external magnetic field. The optimization proceeds in stages: the dressed states of the drift Hamiltonian structure the target, an initial shooting optimization is performed in the rotating-wave frame, and an exact laboratory-frame refinement follows. Benchmarked against Krotov and GRAPE at matched gate infidelity, MAGICARP consistently achieves the lowest energy and a conserved pulse area, concentrates its spectral weight on the gate-relevant transitions, and is the most robust to fluctuations in the exchange coupling; GRAPE independently converges to essentially the same pulse, while Krotov's method pays an order-of-magnitude energy premium. Moreover, a large statistical survey of unselected optimization runs resolves a weak-driving quantum speed limit for two exchange-coupled electron spins: low-amplitude realizations of the two-qubit quantum Fourier transform cease to exist below a critical gate time $T^*$ set by the drift's interaction rate, and the minimum control energy diverges on approach to this limit. The divergence obeys a simple two-parameter area-pole law, $E_2^{\mathrm{law}}(T)=A/T+B/(T-T^*)$, whose first term is the time-optimal area cost and whose second term is a pole at the speed limit.

• The paper presents a novel drift-aware extension of MAGICARP that produces minimal-energy pulses by globally parameterizing control variables.
• Methodologically, it employs a two-stage optimization pipeline that first identifies actionable transitions in a drift-dressed frame and then refines them in the laboratory frame.
• It demonstrates clear advantages over GRAPE and Krotov methods by achieving superior spectral concentration, energy efficiency, and robustness to coupling errors.

Probing Quantum Speed Limits with Drift-Aware Shooting Methods

Overview of Drift-Aware MAGICARP for Quantum Optimal Control

The paper presents a significant advance in quantum optimal control (QOC) by extending the MAGICARP shooting method to closed quantum systems with constant drift Hamiltonians, specifically exchange-coupled spin qubits subjected to external magnetic fields. MAGICARP, inspired by Pontryagin's maximum principle (PMP), structures the control search space via transport of a finite-dimensional anti-Hermitian generator along the quantum trajectory, ensuring the resulting control pulses are inherently smooth and energy-efficient. Unlike GRAPE and Krotov’s methods, which treat each discretized pulse amplitude as an independent parameter and consequently generate broadband, energy-intensive waveforms, MAGICARP parametrizes the pulse globally through a small set of physically interpretable variables, yielding minimal-energy solutions with spectral concentration.

The main technical contribution is embedding MAGICARP in a two-stage optimization pipeline: an initial stage in the drift-dressed rotating-wave frame identifies actionable transition quadratures, followed by laboratory-frame refinement with exact dynamics propagation. This addresses the non-perturbative influence of drift, making the approach suitable for platforms where drift dominates the dynamics (e.g., semiconductor/surface spin qubits, dispersive transmons, NV centers, dipolar molecular qubits).

Benchmarking and Comparative Analysis

Fair-Halting Protocol and Methodological Distinctions

The benchmarking protocol is rigorous: all methods (MAGICARP, Krotov, GRAPE) act on identical models and targets, halt as soon as the independently verified gate infidelity crosses a threshold ($1-F \leq 10^{-5}$ or $10^{-3}$), and the minimal-energy pulse is selected among restarts. This eliminates ambiguity from stopping rules and isolates control cost metrics (pulse energy, area, spectral selectivity, robustness) at matched fidelity.

Spectral Concentration and Energy Efficiency

At matched infidelity, MAGICARP and (zero-initialized) GRAPE converge on spectrally concentrated pulses, with almost all spectral power allocated to gate-relevant dressed transitions; off-resonant leakage is suppressed structurally (MAGICARP) or weakly (GRAPE). Krotov’s method, in contrast, consistently produces broadband solutions — up to a 57x energy penalty, non-conserved area, and significant spectral weight outside gate-relevant bands, especially at shorter gate durations. This distinction persists across different coupling regimes and gates (NOT2, dressed QFT), demonstrating that the structurally minimal-energy solution is intrinsic to the control landscape (not merely an artifact of the parameterization).

Robustness to Coupling Errors

Minimum-energy pulses from MAGICARP and GRAPE exhibit superior robustness to exchange-coupling errors, degrading more slowly in fidelity under perturbations in $J$ compared to the high-energy, broadband Krotov solutions. This is critical for realistic implementations, given that physical systems invariably deviate from nominal parameters. Robustness is physically rooted: resonant, frequency-encoded pulses are less sensitive to drift parameter uncertainty, whereas high-amplitude solutions are fragile.

Weak-Driving Quantum Speed Limit and Control Landscape

Statistical Survey and Onset Mapping

A 9600-run optimization sweep using drift-aware MAGICARP exposes the intrinsic weak-driving quantum speed limit for the dressed two-qubit QFT. Below a critical gate time $T^*$ (set by the drift's interaction rate), no low-amplitude solution exists. The minimum attainable control energy diverges as $T \to T^*$, following the area-pole law:

$E_\text{law}(T) = \dfrac{A}{T} + \dfrac{B}{T-T^*}$

where $A$ reflects the conserved pulse area, and $B$ quantifies the pole at the speed limit. Numerical results pinpoint $T^* \approx 13.48$ ns, coinciding with the derived single-axis interaction bound from the drift Hamiltonian.

Control Cost Scaling and Landscape Structure

Pulse area $E_1$ is conserved across gate times — fixed by the Rabi rotation angles for the gate — while energy $10^{-3}$0 and peak amplitude rise as $10^{-3}$1, consistent with geodesic/time-optimal control theory. GRAPE independently rediscovers the minimum-energy floor, confirming the universality of the optimal pulse across methods. Stochastic solver outcomes show basin sensitivity but also demonstrate stringent separation between high-fidelity, low-energy solutions and failed, high-energy ones. The amplitude bound is a safety cap: it constrains catastrophic failure without sacrificing optimality.

Practical and Theoretical Implications

Experimental Compatibility

MAGICARP produces pulses that are smooth, spectrally concentrated, and experimentally feasible with finite bandwidth and amplitude constraints. Structural suppression of spectator transitions and minimal energy cost makes pulses implementable on physical devices, minimizing heat load and decoherence risk.

Quantum Speed Limit Characterization

The methodology offers a clean, operational probe of quantum speed limits in the weak-driving regime, mapping boundaries dictated by the drift Hamiltonian rather than absolute controllability. The onset, scaling law, and pole at $10^{-3}$2 are theoretically relevant for understanding limits of gate synthesis in drift-dominated platforms.

Future Directions

The approach is currently restricted to closed-system dynamics. Extension to open systems via stochastic averaging or Lindblad evolution will be essential to address realistic decoherence and noise. Robust ensemble optimization for parameter uncertainty (e.g., exchange coupling) can further enhance practical robustness. Generalization to multi-channel control and higher-dimensional registers (qudits) is a natural trajectory, leveraging structural advantages of the shooting parametrization (Etienney et al., 23 Mar 2026).

Conclusion

The drift-aware extension of MAGICARP offers a principled, efficient solution to quantum optimal control in systems with constant drift, producing minimal-energy pulses concentrated on relevant transitions. Rigorous benchmarking against standard methods reveals its energy efficiency, spectral parsimony, and robustness, and establishes its utility as an instrument for exploring quantum speed limits in the weak-driving regime. The method's structural discipline is both a practical and theoretical asset, with clear routes for extension to open-system dynamics, ensemble robustness, and higher-dimensional control landscapes.