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arXiv:2403.15785 · Quantum Science and Technology

Optimal control of spin qudits subject to decoherence using amplitude-and-frequency-constrained pulses

Alonso Hernández-Antón, Fernando M. Luis, Alberto Castro

Published Mar 11, 2025• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/2058-9565/adbf44

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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.
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.
The control of quantum dynamics via specially tailored laser pulses is a long-standing goal in physics and chemistry. Partly, this dream has come true, as sophisticated pulse-shaping experiments allow us to coherently control product ratios of chemical reactions. The theoretical design of the laser pulse to transfer an initial state to a given final state can be achieved with the help of quantum optimal control theory (QOCT). This tutorial provides an introduction to QOCT. It shows how the control equations defining such an optimal pulse follow from the variation of a properly defined functional. We explain the most successful schemes to solve these control equations and show how to incorporate additional constraints in the pulse design. The algorithms are then applied to simple quantum systems and the obtained pulses are analysed. Besides the traditional final-time control methods, the tutorial also presents an algorithm and an example to handle time-dependent control targets.
In the context of molecular quantum computation the optimal control theory (OCT) is used to obtain shaped laser pulses for high-fidelity control of vibrational qubits. Optimization is done in time domain and the OCT algorithm varies values of electric field in each time step independently, tuning hundreds of thousands of parameters to find one optimal solution. Such flexibility is not available in experiments, where pulse shaping is done in frequency domain and the number of "tuning knobs" is much smaller. The question of possible experimental interpretations of theoretically found OCT solutions arises. In this work we analyze very accurate optimal pulse that we obtained for implementing quantum gate CNOT for the two-qubit system encoded into the exited vibrational states of thiophosgene molecule. Next, we try to alter this pulse by reducing the number of available frequency channels and intentionally introducing systematic and random errors (in frequency domain, by modifying the values of amplitudes and phases of different frequency components). We conclude that a very limited number of frequency components (only 32 in the model of thiophosgene) are really necessary for accurate control of the vibrational two-qubit system, and such pulses can be readily constructed using OCT. If the amplitude and phase errors of different frequency components do not exceed ±3% of the optimal values, one can still achieve accurate transformations of the vibrational two-qubit system, with gate fidelity of CNOT exceeding 0.99.
Most quantum processors requires pulse sequences for controlling quantum states. Here we present an alternative algorithm for computing an optimal pulse sequence so as to perform a specific task, being an implementation of a quantum gate or a quantum-state preparation. In our method, we reduce greatly the number of parameters to be fitted by using a limited number of functions as the modulations for the amplitude and phase of the radio-frequency pulses. Unlike other algorithms, ours uses some approximations to make it fast and scalable, and uses some efficient methods to obtain pulses that are robust to common types of experimental errors. We demonstrate the success of the proposed algorithm by performing several real experiments for 4-, 7-, and 12-quantum-bit systems using NMR. In addition, we show the efficiency of the algorithm, finding pulses for controlling with good fidelity the quantum states of spins in fictional square two-dimensional lattices containing 16, 36, and 100 qubits.
We address the experimentally relevant problem of robust mitigation of dephasing noise acting on a qubit. We first present an extension of the method of Kuopanportti et al. [Phys. Rev. A 77, 032334 (2008)] for representing $1/{\ensuremath{\omega}}^{\ensuremath{\alpha}}$ noise to the efficient representation of arbitrary Markovian noise. We then add qubit control pulses to enable the design of numerically optimized, two-dimensional, bounded amplitude control functions capable of decoupling the qubit from the dephasing effects of a broad variety of Markovian noise spectral densities during one- and two-qubit quantum operations. We illustrate the method with development of numerically optimized control pulse sequences that minimize decoherence due to a combination of $1/\ensuremath{\omega}$ and constant-offset noise sources. Comparison with the performance of standard dynamical decoupling protocols shows that the numerically optimized pulse sequences are considerably more robust with respect to zero-frequency noise. Application to the mitigation of dephasing noise on spin qubits in silicon indicates that high-fidelity quantum gates may, in principle, be implemented for such qubits with the assistance of current pulse-generation technology.