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arXiv:2606.22212

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

Denis Janković, Saba Taherpour, Paul-Louis Etienney, Paul-Antoine Hervieux, Christoph Wolf

quant-phmath-phphysics.comp-phSubmitted Jun 20, 2026• Announced Jun 23, 2026• 0 citations• Revised Sep 21, 2026 (v2)
v1 · Jun 20, 2026v2 · Sep 21, 2026
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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 black-box Krotov meets the same error at an order-of-magnitude energy premium. This method-independence is what qualifies the bounded solver as a measurement instrument, and the central result follows: 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, and the minimum control energy diverges on approach to this limit. The divergence obeys a simple two-parameter area--pole law, $E_2^{\mathrm{law}}(T)=A/T+B/(T-T^*)$.

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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 black-box Krotov meets the same error at an order-of-magnitude energy premium. This method-independence is what qualifies the bounded solver as a measurement instrument, and the central result follows: 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, and the minimum control energy diverges on approach to this limit. The divergence obeys a simple two-parameter area--pole law, E2law(T)=A/T+B/(T−T∗)E_2^{\mathrm{law}}(T)=A/T+B/(T-T^*)E2law​(T)=A/T+B/(T−T∗).
We introduce the MAGICARP algorithm, a numerical optimization method for quantum optimal control problems that combines the structure provided by Pontryagin's Maximum Principle (PMP) and the robustness of gradient ascent techniques, such as GRAPE. MAGICARP is formulated as a "shooting technique", aiming to determine the appropriate initial adjoint momentum to realize a target quantum gate. This method naturally incorporates time and energy optimal constraints through a PMP-informed pulse structure. We demonstrate MAGICARP's effectiveness through illustrative numerical examples, comparing its performance to GRAPE and highlighting its advantages in specific scenarios.
High-fidelity quantum control is a cornerstone of scalable quantum technologies. We introduce a shooting-based optimization framework that generates smooth, experimentally realistic control pulses for implementing quantum gates in discrete quantum systems. Through numerical simulations on realistic architectures inspired by single-molecule magnets, we demonstrate that our method efficiently decomposes target quantum operations into electric pulse sequences while outperforming the widely used GRAPE algorithm.
Quantum optimal control can produce fast and robust multi-qubit gates, but experimentally calibrating the resulting high-dimensional waveforms remains challenging because direct searches over large parameter spaces converge slowly. Building on the low-rank structure of quantum-control landscapes, we develop and benchmark a Hessian-based calibration method for optimal-control gates. The method identifies the few waveform directions that affect fidelity to leading order, with the number of directions set by the accessible leakage and coherent error channels, and optimizes only within this principal space using closed-loop experimental feedback. We apply this approach to an amplitude-robust controlled-Z gate on metastable-state 171Yb nuclear-spin qubits. Experimentally, we verify the predicted Hessian-sensitive directions and demonstrate rapid convergence of the optimization protocol. The optimized gate reaches a raw fidelity of 0.9959(2), increasing to 0.99902(7) after postselection on no detected loss, and the performance is essentially unchanged under laser-power variations of up to 20%. We further show that the same fidelity Hessian directions can correct certain Hamiltonian parameter errors. These results establish low-rank Hessian optimization as an efficient and physically motivated calibration strategy for high-dimensional optimal-control gates, which is broadly applicable to many qubit types.
Constructing high-fidelity control fields that are robust to control, system, and/or surrounding environment uncertainties is a crucial objective for quantum information processing. Using the two-state Landau-Zener model for illustrative simulations of a controlled qubit, we generate optimal controls for $\ensuremath{\pi}/2$ and $\ensuremath{\pi}$ pulses and investigate their inherent robustness to uncertainty in the magnitude of the drift Hamiltonian. Next, we construct a quantum-control protocol to improve system-drift robustness by combining environment-decoupling pulse criteria and optimal control theory for unitary operations. By perturbatively expanding the unitary time-evolution operator for an open quantum system, previous analysis of environment-decoupling control pulses has calculated explicit control-field criteria to suppress environment-induced errors up to (but not including) third order from $\ensuremath{\pi}/2$ and $\ensuremath{\pi}$ pulses. We systematically integrate this criteria with optimal control theory, incorporating an estimate of the uncertain parameter to produce improvements in gate fidelity and robustness, demonstrated via a numerical example based on double quantum dot qubits. For the qubit model used in this work, postfacto analysis of the resulting controls suggests that realistic control-field fluctuations and noise may contribute just as significantly to gate errors as system and environment fluctuations.
The quantum control of spin-correlated radical pairs (SCRPs) holds promise for the targeted manipulation of magnetic field effects (MFEs), with potential applications ranging from the design of noise-resilient quantum information processors to genetically encodable quantum sensors. However, achieving precise handles over the intricate interplay between coherent electron spin dynamics and incoherent relaxation processes in photo-excited radical-pair reactions requires tractable approaches for numerically obtaining controls for large, complex open quantum systems. To that end, we realise optimal control of SCRPs in the low-magnetic-field regime, through first generalising the gradient-ascent pulse engineering (GRAPE) paradigm to allow for optimising time-integral reaction yields for coherent control of SCRP spin dynamics. We then explore how interradical interactions which are unavoidable in a biophysical setting, such as electron-electron dipolar and exchange couplings, alongside driving resulting from intrinsic structural motion, can push the precision of the proposed radical pair quantum compass in cryptochrome closer to approaching the quantum Cram\'{e}r-Rao bound. Remarkably, this approach to optimality is amplified under environmental noise and persists with increasing complexity of the spin system, suggesting that perturbations inherent to such natural systems have enabled them to operate closer to the quantum limit to more effectively extract information from the weak geomagnetic field. Bolstered by these findings, we furthermore demonstrate how a control engineering approach based on the Pontryagin Maximum Principle (PMP) can offer a more computationally feasible route towards realising not only coherent, but also incoherent control, of noisy SCRP open-system spin dynamics. This raises prospects for experimental realisation of spin control of actual radical-pair systems in ambient magnetic fields, by suppressing or boosting radical reaction yields using purpose-specific radio-frequency wave forms, paving the way for inspiring more noise-robust quantum magnetometry via chemical reaction-yield-dependence, and potentially applications of quantum control to physiologically relevant radical-pair reaction intermediates.
We address a control-constrained optimal control problem pertaining to the transformation of quantum states. Our objective is to steer a quantum system from an initial state to a desired target state while adhering to the principles of the Liouville-von Neumann equation. To achieve this, we introduce a cost functional that balances the dual goals of fidelity maximization and energy consumption minimization. We derive optimality conditions in the form of the Pontryagin Maximum Principle (PMP) for the matrix-valued dynamics associated with this problem. Subsequently, we present a time discretized computational scheme designed to solve the optimal control problem. This computational scheme is rooted in an indirect method grounded in the PMP, showcasing its versatility and efficacy. To illustrate the practicality and applicability of our methodology, we employ it to address the case of a spin ½ particle subjected to interaction with a magnetic field. Our findings shed light on the potential of this approach to tackle complex quantum control scenarios and contribute to the broader field of quantum state transformations.