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Physical Review Applied

Optimal Control of Molecular Spin Qudits

Alberto Castro, Adrián García Carrizo, Sebastián Roca-Jerat, David Zueco, Fernando M. Luis

Published Jun 14, 2022• 46 citations
Publisher

Abstract

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 ${\mathrm{Gd}}^{3+}$ ion. Its eight accessible levels (corresponding to a total spin $S=7/2$) 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.

DOI
10.1103/physrevapplied.17.064028

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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.
Abstract Quantum optimal control theory (QOCT) can be used to design the shape of electromagnetic pulses that implement operations on quantum devices. By using non-trivially shaped waveforms, gates can be made significantly faster than those built by concatenating monochromatic pulses. Recently, we applied this idea to the control of molecular spin qudits modeled with Schrödinger’s equation and showed it can speed up operations, helping mitigate the effects of decoherence (Castro et al 2022 Phys. Rev. Appl. 17 064028). However, short gate times require large optimal pulse amplitudes, which may not be experimentally accessible. Introducing bounds to the amplitudes then unavoidably leads to longer operation times, for which decoherence can no longer be neglected. Here, we study how to improve this procedure by applying QOCT on top of Lindblad’s equation, to design control pulses accounting for decoherence already in the optimization process. We define the control signal in terms of generic parameters, which permits the introduction of bounds and constraints. This is convenient, as amplitude and frequency limitations are inherent to waveform generators. The pulses that we obtain consistently enhance operation fidelities compared to those achieved with the optimization based on Schrödinger’s equation, demonstrating the flexibility and robustness of our method. The improvement is larger the shorter the spin coherence time T 2.
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.
Abstract Molecular Nanomagnets may enable the implementation of qudit-based quantum error-correction codes which exploit the many spin levels naturally embedded in a single molecule, a promising step towards scalable quantum processors. To fully realize the potential of this approach, a microscopic understanding of the errors corrupting the quantum information encoded in a molecular qudit is essential, together with the development of tailor-made quantum error correction strategies. We address these central points by first studying dephasing effects on the molecular spin qudit produced by the interaction with surrounding nuclear spins, which are the dominant source of errors at low temperatures. Numerical quantum error correction codes are then constructed, by means of a systematic optimization procedure based on simulations of the coupled system-bath dynamics, that provide a striking enhancement of the coherence time of the molecular computational unit. The sequence of pulses needed for the experimental implementation of the codes is finally proposed.
Molecular nanomagnets are quantum spin systems potentially serving as qudits for future quantum technologies thanks to their many accessible low-energy states. At low temperatures, the primary source of error in these systems is pure dephasing, caused by their interactions with the bath of surrounding nuclear spins degrees of freedom. Most importantly, as the system’s dimensionality grows going from qubits to qudits, the control and mitigation of decoherence become more challenging. Here, we analyze the characteristics of pure dephasing in molecular qudits under spin-echo sequences. We use a realistic description of their interaction with the bath, whose non-Markovian dynamics is accurately computed by the cluster correlation expansion technique. First, we show that the differences in the expectation values of the local spin operators on the eigenstates of the qudit are the source of decoherence in these systems. Indeed, we demonstrate that this is a necessary and sufficient condition to prevent the decay of coherence with time, also introducing a parameter to quantify the deviation from such ideal condition. We illustrate this with two paradigmatic systems: a single giant spin and a composite antiferromagnetic spin system. We then advance a proposal for optimized nanomagnets, identifying key ingredients for engineering robust qudits for quantum technologies.
The implementation of a universal quantum processor still poses fundamental issues related to error mitigation and correction, which demand to investigate also platforms and computing schemes alternative to the main stream. A possibility is offered by employing multi-level logical units (qudits), naturally provided by molecular spins. Here we present the blueprint of a Molecular Spin Quantum Processor consisting of single Molecular Nanomagnets, acting as qudits, placed within superconducting resonators adapted to the size and interactions of these molecules to achieve a strong single spin to photon coupling. We show how to implement a universal set of gates in such a platform and to readout the final qudit state. Single-qudit unitaries (potentially embedding multiple qubits) are implemented by fast classical drives, while a novel scheme is introduced to obtain two-qubit gates via resonant photon exchange. The latter is compared to the dispersive approach, finding in general a significant improvement. The performance of the platform is assessed by realistic numerical simulations of gate sequences, such as Deutsch-Josza and quantum simulation algorithms. The very good results demonstrate the feasibility of the molecular route towards a universal quantum processor.