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Frontiers of Physics

Robust optimization of two-qubit gates using the GRAPE algorithm in nitrogen−vacancy centers

Jun-Tong Ding, Xinyu Zhao, Ye‐Hong Chen, Zhi‐Cheng Shi, Yan Xia

Published Jan 1, 2026• 0 citations
Publisher

Abstract

We propose a scheme for optimizing two-qubit gate operations in a decoherence-free subspace (DFS) using the gradient ascent pulse engineering (GRAPE) algorithm, where the system consists of nitrogen-vacancy (NV) centers in diamond coupled to a whispering-gallery-mode microresonator. The logical qubits in this scheme are encoded in the DFS, which effectively avoids collective decoherence and provides a long coherence time for high-fidelity quantum operations. The GRAPE algorithm is used to design and optimize the control pulse sequences required for the target operation, thereby compensating for common errors and ensuring that the quantum gate maintains high fidelity under non-ideal conditions. As examples, we successfully implement high-fidelity and robust controlled-phase and SWAP gates. Numerical simulations demonstrate the effectiveness and superiority of the proposed scheme. The strategy of combining encoding protection with optimal control shown in this work can be extended to implement universal quantum gates in complex quantum systems.

DOI
10.15302/frontphys.2026.103202

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We propose a scheme for optimizing two-qubit gate operations in a decoherence-free subspace (DFS) using the gradient ascent pulse engineering (GRAPE) algorithm, where the system consists of nitrogen-vacancy (NV) centers in diamond coupled to a whispering-gallery-mode microresonator. The logical qubits in this scheme are encoded in the DFS, which effectively avoids collective decoherence and provides a long coherence time for high-fidelity quantum operations. The GRAPE algorithm is used to design and optimize the control pulse sequences required for the target operation, thereby compensating for common errors and ensuring that the quantum gate maintains high fidelity under non-ideal conditions. As examples, we successfully implement high-fidelity and robust controlled-phase and SWAP gates. Numerical simulations demonstrate the effectiveness and superiority of the proposed scheme. The strategy of combining encoding protection with optimal control shown in this work can be extended to implement universal quantum gates in complex quantum systems.
In the paper, we present schemes for implementing multi-qubit quantum gates in quantum networks based on the nitrogen-vacancy (NV) centers in diamond. In our schemes, NV centers coupled to the whispering-gallery modes of microtoroidal resonators serve as quantum registers to store quantum information. By coding the qubits of NV centers into decoherence-free subspaces, quantum information encoded in logical qubits is protected from collective dephasing, and quantum controlled-NOT and Toffoli gates between logical qubits from the same or different quantum nodes can be implemented. Compared with the previous works, our schemes are simpler and reduce the consumption of resources. Furthermore, the modularized design of multi-qubit quantum gates can be extended to other quantum platforms, which may lead to more efficient construction of quantum networks for distributed quantum computation and quantum communication.
Designing optimal control for multiparameter quantum sensing is essential for approaching the ultimate precision limits. However, analytical solutions are generally available only for simple systems, while realistic scenarios often involve coupled qubits and time-dependent Hamiltonians. Here we numerically investigate optimal control of a two-qubit sensor-ancilla system coupled via an Ising term using Gradient Ascent Pulse Engineering (GRAPE) to minimize the objective function. By seeding the optimization recursively with solutions obtained for smaller coupling strengths and selecting a suitable initial guess, we achieve robust convergence and high precision across a wide range of interaction strengths and field configurations. The proposed approach offers a practical route toward high-sensitivity, robust multiparameter magnetometry and it is applicable to solid-state quantum sensors such as nitrogen-vacancy (NV) centers in realistic experimental settings.
This thesis is concerned with the control of quantum systems.Given a Hamiltonian model of a quantum system, we are interested in finding controls-typically shaped electromagnetic pulses-that steer the evolution of the system toward a desired target operation.For this we employ a numerical optimisation method known as the GRAPE algorithm.For particular experimental systems, we design control schemes that respect constraints of robustness and addressability, and are within the reach of the experimental hardware.A general procedure is given for specifying a Hamiltonian model of a driven N -level system and converting it to an appropriate rotating frame.This is then applied together with the numerical algorithm to design improved schemes for two different systems, where laser fields manipulate orbital and hyperfine states of Pr 3+ and Rb.The generation of cluster states in Ising-coupled systems is also studied.We find that, in the ideal case, the solution of evolving only under the coupling Hamiltonian is not time-optimal.This surprising result is in contrast to the known cases for unitary gates.For a symmetrised three-qubit example, we provide a geometrical interpretation of this.Numerically optimised control schemes are then developed for a nonideal coupling topology, modelling an experimental configuration of trapped ions.Controls for the implementation of the two-qubit Deutsch and Grover algorithms are designed for a pair of 13 C nuclear spins at a nitrogen vacancy center in diamond.These implementations are robust to experimental errors, and found to be reproduced with high accuracy on a VFG-150 pulse generator.We also consider two-qubit gate synthesis in a system of superconducting qubits coupled by mii xii CONTENTS C.4 Examples of time-optimal gates . . . . . . . .
Numerical optimizations of the microwave and radio frequency control pulses for the nitrogen vacancy electron-nuclear spin register were performed by means of a gradient ascent pulse engineering method (GRAPE). Three examples of spin control — specific quantum state preparation and the implementation of a controlled-NOT and unconditional - gate have been optimized through parallel computation. The optimized control pulses reach fidelities of F = 0:99944, F = 0:77929 and F = 0:99143 respectively, whereas simulation of Rabi oscillation based control yields corresponding results of F = 0:96325, F = 0:21884 and F = 0:42702. The fixed step size in the GRAPE algorithm imposes a tradeoff between monotone and fast convergence of optimized control pulses and the unspecific performance function may lead to erroneous optimizations. These complications require future research to be resolved. Furthermore, implementing the effect of decoherence yields no significant average fidelity improvement, but has a positive effect on reducing the spread of final density states for the case of quantum state preparation.
In this paper, we demonstrate how gradient ascent pulse engineering (GRAPE) optimal control methods can be implemented on donor electron spin qubits in semiconductors with an architecture complementary to the original Kane’s proposal. We focus on the high fidelity controlled‐NOT (CNOT) gate and we explicitly find the digitized control sequences for a controlled‐NOT gate by optimizing its fidelity using the effective, reduced donor electron spin Hamiltonian with external controls over the hyperfine A and exchange J interactions. We then simulate the CNOT‐gate sequence with the full spin Hamiltonian and find that it has an error of 10−6 that is below the error threshold of 10−4 required for fault‐tolerant quantum computation. Also the CNOT gate operation time of 100 ns is 3 times faster than 297 ns of the proposed global control scheme.
Nitrogen-vacancy (NV) center ensembles have the potential to improve a wide range of applications, including nuclear magnetic resonance spectroscopy at the microscale and nanoscale, wide-field magnetometry, and hyperpolarization of nuclear spins via the transfer of optically induced NV polarization to nearby nuclear spin clusters. These NV ensembles can be coherently manipulated with microwave cavities, that deliver strong and homogeneous drivings over large volumes. However, the pulse shaping for microwave cavities presents the challenge that the external controls and intracavity field amplitudes are not identical, leading to adverse effects on the accuracy of operations on the NV ensemble. In this paper, we introduce a method based on gradient ascent pulse engineering (GRAPE) to optimize external controls, resulting in robust pulses within the cavity while minimizing the effects of cavity ringings. The effectiveness of the method is demonstrated by designing both π and π / 2 pulses. These optimized controls are then integrated into a PulsePol sequence, where numerical simulations reveal a resilience to detunings five times larger than those tolerated by the sequence constructed using standard controls.