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Physical Review A

Time-optimal control of a GdW 30 molecular spin qudit

Marc Alsina, Ibério de P. R. Moreira, Guillermo Albareda, Josep María Bofill

Published Mar 31, 2026• 0 citations
Publisher

Abstract

Quantum systems with more than two levels, known as qudits, offer enhanced information density and potentially more efficient quantum operations compared with traditional qubits. Among the various physical platforms under investigation, single-molecule magnets (SMMs) stand out due to their intrinsic magnetic anisotropy, long spin-relaxation times, and the natural presence of multiple accessible spin states. In this work, we explore two distinct control strategies for implementing state-to-state transitions within the multilevel structure of SMMs, viz., a resonant magnetic-field manipulation ( π -pulse control) and the so-called quantum Zermelo navigation approach, which provides the time-optimal Hamiltonian that transforms a given initial state to a given target state. We present a comparative analysis of these two methods, evaluating their operational efficiency and feasibility for experimental realization. Our results show that, while π -pulse control is limited by strict timing constraints, the Zermelo method faces significant challenges in terms of its physical implementation.

DOI
10.1103/c7yr-jrs4

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•Mar 1, 2014•1 citation
Quantum systems with more than two levels, known as qudits, offer enhanced information density and potentially more efficient quantum operations compared with traditional qubits. Among the various physical platforms under investigation, single-molecule magnets (SMMs) stand out due to their intrinsic magnetic anisotropy, long spin-relaxation times, and the natural presence of multiple accessible spin states. In this work, we explore two distinct control strategies for implementing state-to-state transitions within the multilevel structure of SMMs, viz., a resonant magnetic-field manipulation ( π -pulse control) and the so-called quantum Zermelo navigation approach, which provides the time-optimal Hamiltonian that transforms a given initial state to a given target state. We present a comparative analysis of these two methods, evaluating their operational efficiency and feasibility for experimental realization. Our results show that, while π -pulse control is limited by strict timing constraints, the Zermelo method faces significant challenges in terms of its physical implementation.
We demonstrate, numerically, the possibility of manipulating the spin states of molecular nanomagnets with shaped microwave pulses designed with quantum optimal control theory techniques. The state-to-state or full gate transformations can be performed in this way in shorter times than using simple monochromatic resonant pulses. This enhancement in the operation rates can therefore mitigate the effect of decoherence. The optimization protocols and their potential for practical implementations are illustrated by simulations performed for a simple molecular cluster hosting a single Gd3+{\mathrm{Gd}}^{3+}Gd3+ ion. Its eight accessible levels (corresponding to a total spin S=7/2S=7/2S=7/2) allow encoding an eight-level qudit or a system of three coupled qubits. All necessary gates required for universal operation can be obtained with optimal pulses using the intrinsic couplings present in this system. The application of optimal control techniques can facilitate the implementation of quantum technologies based on molecular spin qudits.
We demonstrate, numerically, the possibility of manipulating the spin states of molecular nanomagnets with shaped microwave pulses designed with quantum optimal control theory techniques. The state-to-state or full gate transformations can be performed in this way in shorter times than using simple monochromatic resonant pulses. This enhancement in the operation rates can therefore mitigate the effect of decoherence. The optimization protocols and their potential for practical implementations are illustrated by simulations performed for a simple molecular cluster hosting a single Gd3+^{3+}3+ ion. Its eight accessible levels (corresponding to a total spin S=7/2S=7/2S=7/2) allow encoding an 888-level qudit or a system of three coupled qubits. All necessary gates required for universal operation can be obtained with optimal pulses using the intrinsic couplings present in this system. The application of optimal control techniques can facilitate the implementation of quantum technologies based on molecular spin qudits.
We use a specific geometric method to determine speed limits for the implementation of quantum gates in controlled quantum systems that have a specific class of constrained control functions. We achieve this by applying a recent theorem of Shen, which provides a connection between time optimal navigation on Riemannian manifolds and the geodesics of a certain Finsler metric of Randers type. We use the lengths of these geodesics to derive the optimal implementation times (under the assumption of constant control fields) for an arbitrary quantum operation (on a finite dimensional Hilbert space), and explicitly calculate the result for the case of a controlled single spin system in a magnetic field and a swap gate in a Heisenberg spin chain.
We study photon-mediated interactions between molecular spin qudits in the dispersive regime of operation. We derive from a microscopic model the effective interaction between molecular spins, including their crystal field anisotropy (i.e., the presence of nonlinear spin terms) and their multilevel structure. Finally, we calculate the long-time dynamics for a pair of interacting molecular spins using the method of multiple-scale analysis. This allows us to find a set of two-qudit gates that can be realized for a specific choice of molecular spins and to determine the time required for their implementation. Our results are relevant for the implementation of logical gates in general systems of qudits with unequally spaced levels or to determine an adequate computational subspace to encode and process the information.
Zero-field nuclear magnetic resonance (NMR) has been recently developed as a complementary tool for nondestructive structural investigations of matter. However, precise and efficient control in such systems is still at the beginning. Based on the controllability of spin systems, we theoretically study the issue of quantum control in zero-field NMR by using a numerical optimization method based on the gradient ascent pulse engineering (GRAPE) algorithm. A set of quantum gates, including single-qubit and two-qubit gates, are achieved as examples. In particular, we show that homonuclear spins can be individually manipulated in zero-field NMR by the GRAPE method, which extends the ability to manipulate spin systems at zero magnetic field. Quantum control realized by the numerical GRAPE method features the properties of high fidelity, robustness, and hardware-friendly characteristics. High-fidelity quantum control in zero-field NMR systems might provide promising applications in material science, chemical analysis, quantum information processing, and fundamental physics.
During the last ten years, basic quantum operations on a single electronic spins in the nitrogen-vacancy defect center in diamond have been a rich field of study. Quantum operations coherently manipulating the spin of the nitrogen-vacancy center have become promising candidates for easy to manipulate qubits. However in order to implement these qubits in quantum computers robust high-fidelity control is needed. Building on an existing confocal microscopy setup, pulsed and quadrature-amplitude modulated microwave signals were used to realize coherent spin manipulation measurements. Especially designed Smooth Optimal Control pulses (robust against detuning and different control amplitudes) were used to develop measurement schemes resulting in the aforementioned increase of fidelity. The first part of the thesis describes the implementation of these pulses into the existing experimental setup and the calibration work necessary prior to measurements, due to imperfections of devices in the microwave chain. The second part is dedicated to the analysis of the pulses applied to a single spin by comparing the experimental results to theoretical simulations. This was done not only for one resonant spin with accurate control amplitude, but also for a range of different detunings and amplitudes. A more thorough analysis of the pulses was then achieved by using state/process tomography, resulting in a complete description of the process induced by these optimized pulses. Finally, after verifying the effect of the pulses, measurement schemes for sensing alternating magnetic fields were implemented, resulting in measurements with improved sensitivity for magnetic fields.